UNIVERSITY  OF  CALIFORNIA 
AT   LOS  ANGELES 


EFFECTS   OF  WINDS  AND  OF 

BAROMETRIC  PRESSURES 

ON  THE  GREAT  LAKES 


BY 

JOHN  F.  HAYFORD 

RESEARCH  ASSOCIATE,  CARNEGIE  INSTITUTION  OF  WASHINGTON 


PUBLISHED  BY  THE  CARNEGIE  INSTITUTION  OP  WASHINGTON 
WASHINGTON,  OCTOBER,  1922 


CARNEGIE  INSTITUTION  OF  WASHINGTON 
-  PUBLICATION  No.  317 


THE    RUMFORD    PRESS 
CONCORD 


fcng  inhering 
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CONTENTS. 


• PAGES 

General  introduction 1-4 

Data  used  and  acknowledgments 4-5 

Methods  of  this  investigation 6-8 

Outcome  of  the  investigation 8-9 

Order  of  presentation 9-10 

Theoretical  basis  for  barometric-observation  equations 10-12 

Barometric  effects  in  terms  of  barometric  pressures 12-15 

Expression  of  barometric  gradients 15-16 

Proportionality  factors  for  barometric  effects 16-18 

Form  of  observation  equations  for  barometric  effects 18-20 

Lag  in  barometric  effects 20-23 

Values  of  Cw  and  Cn 23 

Assumed  uniform  change  in  barometric  gradient 23-24 

Example  of  observation  equations  for  barometric  effect 24-26 

Barometric  terms  in  observation  equations 26-27 

Corrected  change  of  elevation  used  in  observation  equations 28-29 

Example  of  normal  equations  for  barometric  effects 29-30 

Example  of  substitution  in  observation  equations  for  barometric  effects . . .  30-31 

The  five  final  solutions  for  barometric  effects 31-32 

Computation  of  hourly  barometric  effects 32-36 

Computation  of  daily  barometric  effects 36-39 

Theoretical  basis  for  wind  observation  equations 39-40 

Relation  between  depth  and  slope  produced  by  wind 40-43 

Derivation  of  2X 43-44 

Example  of  computation  of  2X 44-48 

Determination  of  position  of  nodal  line 48-51 

Positions  of  various  nodal  lines 51-52 

Values  of  Sx 52-54 

The  wind  exponent 54-55 

Example  of  observation  equations  for  wind  effects 55-61 

Examples  of  normal  equations  for  wind  effects . 61 

The  four  final  wind  solutions , 61-63 

Computation  of  wind  effects 63-64 

Accuracy  of  computed  barometric  effects 64-65 

The  rejection  rule 65-66 

Rule  for  combining  observation  equations 66-67 

Discrepancies  between  pairs  of  values 67-69 

Study  of  proportionality  factors 69-72 

Conclusions  on  accuracy  of  computed  barometric  effects 72 

Accuracy  of  computed  wind  effects 72-77 

Accuracy  of  corrected  elevations  of  water  surface 78-79 

Mean  elevations  of  Lake  Erie 79-100 

Observed  and  corrected  daily  elevations  at  Buffalo 80-83 

Observed  and  corrected  daily  elevations  at  Cleveland 84-87 

Observed  and  corrected  daily  elevations  at  Milwaukee 88-90 

Observed  and  corrected  daily  elevations  at  Harbor  Beach 91-93 

Observed  and  corrected  daily  elevations  at  Mackinaw 94-96 

iii 


435704 


iv  CONTENTS. 

PAGES 

Observed  and  corrected  elevations  for  the  whole  surface  of  Lake  Erie 97-100 

Mean  elevations  of  Lake  Michigan-Huron 100-104 

Observed  and  corrected  elevations  for  the  whole  surface  of  Lake  Michigan- 
Huron 101-103 

Explanations  of  plates  7  to  13 104-105 

Accuracy  as  tested  by  graphs 105-108 

Monthly  and  seasonal  mean  elevations 108 

Probable  errors  and  weights 108-113 

Tides 113 

Seiches 114 

Seiches  at  Buffalo 114-116 

Seiches  at  Cleveland 116-119 

Seiches  at  Strait  of  Mackinac 119-120 

Examples  of  seiches 120-123 

Generalizations 123 

As  to  barometric  effects 123-124 

As  to  wind  effects 124-125 

As  to  seiches 125-128 

As  to  prevailing  conditions  on  the  Great  Lakes 128-129 

Possible  applications  of  results  of  this  investigation 129 

Application  to  a  study  of  laws  of  evaporation 129-130 

Application  to  regulation  of  the  Great  Lakes 130-131 

Application  to  determination  of  mean  sea-level  and  to  precise  leveling 131-132 

Application  to  determination  of  tilting  of  the  Great  Lakes  region 133 


LIST  OF  ILLUSTRATIONS. 

Plate    1.  Isobars  and  contours  on  water  surface  of  Lake  Erie,  August  5,  1910. 

2.  Nodal  lines  and  depths  in  Lake  Erie;  barometric  points  around  the  Great 

Lakes. 

3.  Relation  of  barometric  change  to  rise  of  water. 

4.  Wind  effects,  disturbed  water  surface,  and  currents. 

5.  Nodal  lines  and  depths  in  Lake  Michigan. 

6.  Nodal  lines  and  depths  in  Lake  Huron. 

7.  Elevations  of  water  surface,  Lake  Erie,  June  1-July  15,  1910. 

8.  Elevations  of  water  surface,  Lake  Erie,  July  15-August  28,  1910. 

9.  Elevations  of  water  surface,  Lake  Erie,  August  28-October  11,  1910. 

10.  Elevations  of  water  surface,  Lake  Erie,  October  11-31,  1910. 

11.  Elevations  of  water  surface,  Lake  Michigan-Huron,  June  1-July  19,  1911. 

12.  Elevations  of  water  surface,  Lake  Michigan-Huron,  July  15-August  28, 1911. 

13.  Elevations  of  water  surface,  Lake  Michigan-Huron,  August  28-September 

30,  1911. 

14.  Observed  and  corrected  elevations  of  water  surface,  Buffalo,  August  4-7, 

1910;  barometric  effects  and  wind  effects. 

15.  Observed  and  corrected  elevations  of  water  surface,  Buffalo,  October  21-22, 

1909;  barometric,  effects  and  wind  effects. 

16.  Observed  and  corrected  elevations  of  water  surface,  Buffalo,  October  26-27, 

1910,  and  Cleveland,  October  27,  1910;  also  barometric  effects  and 
wind  effects. 


EFFECTS  OF  WINDS  AND  OF  BARO- 
METRIC PRESSURES  ON  THE 
GREAT  LAKES. 


GENERAL  INTRODUCTION. 

The  investigation  of  wind  effects  and  barometric  effects*  on  the  Great 
Lakes,  which  is  the  subject  of  this  publication,  is  a  part  of  a  much  larger  in- 
vestigation, covering  a  much  broader  field,  which  has  been  in  progress  since 
the  summer  of  1911  except  as  interrupted  by  the  Great  War.  The  ultimate 
object  of  the  larger  investigation  is  to  obtain  a  much  better  formulation  than 
the  engineering  profession  now  has  of  the  laws  governing  the  amount  of 
stream-flow. 

Whenever  it  is  proposed  to  use  the  water  which  is  being  delivered  by  a 
river,  for  the  development  of  power,  for  irrigation,  or  for  the  supply  of  a  city 
or  town,  one  is  confronted  with  the  desirability  of  predicting  the  amount  of 
the  future  flow  of  that  river  and  its  variations.  It  is  necessary  to  know,  as 
a  basis  for  the  best  design  of  the  proposed  works,  not  only  the  mean  flow 
of  the  stream,  but  usually  also  its  maximum  flow,  its  minimum  flow,  and  in 
general  the  characteristic  features  of  its  variation  in  flow.  The  more  ac- 
curately these  things  can  be  predicted  the  better  the  design  of  the  works  may 
be  made  with  respect  to  economy  and  safety.  The  decision,  in  the  first  place, 
as  to  whether  the  proposed  works  are  worth  while  frequently  turns  upon  the 
predicted  future  flow  of  the  stream.  So,  too,  when  it  is  proposed  to  con- 
struct great  engineering  works  for  the  prevention  of  damage  from  floods, 
such  as  those  now  under  construction  in  the  Miami  Conservancy  District, 
for  the  protection  of  Dayton,  Ohio,  and  other  nearby  towns,  one  needs  at 
the  outset  accurate  predictions  of  the  future  flow  in  the  streams  concerned, 
especially  of  the  amount  of  the  flood  flows  and  their  duration.  Any  substan- 
tial improvement  in  the  possibilities  of  accurate  prediction  of  the  future 
flow  of  streams  would  be  of  great  value  to  the  engineers  and  to  humanity. 

It  appeared  to  be  probable  that  progress  toward  this  ultimate  object,  the 
securing  of  a  better  formulation  of  the  laws  governing  the  amount  of  stream- 
flow,  is  conditioned  upon  first  securing  a  better  knowledge  than  is  now 
available  of  the  laws  of  evaporation  from  large  water  surfaces,  such  as  the 
surfaces  of  lakes  and  rivers,  and  from  land  surfaces. 

To  secure  a  better  understanding  of  the  laws  of  evaporation,  it  seemed  to 
be  of  first  importance  to  supplement  the  numerous  and  intensive  studies 

*  Throughout  this  paper  the  effects  of  barometric  pressures  are  termed  barometric 
effects. 

1 


2  EFFECTS   OF   WINDS   AND    OF 

of  evaporation,  which  have  been  made  on  a  small  scale  by  using  evaporation 
pans  with  an  area  of  a  few  square  feet  or  a  few  square  inches,  by  an  investi- 
gation made  on  the  full  scale  of  nature  and  under  natural  conditions.  For 
this  purpose  it  was  proposed  to  consider  each  of  the  Great  Lakes  in  turn  as 
an  evaporation  pan  and  to  evaluate  from  day  to  day  (1)  the  change  of  content, 
(2)  the  income,  and  (3)  the  outgo,  including  evaporation.  There  is  hope 
that  these  evaluations  can  be  made  with  sufficient  accuracy  to  segregate 
that  part  of  the  outgo  which  is  evaporation,  and  to  determine  the  laws  which 
control  the  rate  of  evaporation.  If  that  hope  is  realized,  the  laws  so  deter- 
mined may  be  applied  with  a  confidence  based  upon  the  fact  that  the 
experimental  evidence  is  on  the  full  scale  of  nature  and  under  natural 
conditions. 

The  change  of  content  of  any  one  of  the  Great  Lakes  from  day  to  day  is 
measured  by  the  change  of  elevation  of  the  mean  lake  surface  from  day  to 
day.  The  area  of  the  lake  surface  remains  substantially  constant.  For 
Lake  Erie,  for  example,  the  area  never  varies  by  as  much  as  0.01  part  from 
9,968  square  miles.  If  the  elevation  of  the  mean  surface  of  the  lake  is  ob- 
served to  have  increased  by  0.01  foot,  the  content  of  the  lake  is  known  to 
have  increased  by  the  net  addition  of  a  sheet  of  water  0.01  foot  thick  having 
an  area  of  9,968  square  miles.  Such  a  sheet  contains  about  2,800,000,000 
cubic  feet  and  is  equivalent  to  the  outflow  through  the  Niagara  River  under 
typical  conditions  for  about  3.7  hours — at  210,000  cubic  feet  per  second. 

The  elevation  of  the  surface  of  each  of  the  Great  Lakes  has  been  deter- 
mined at  a  few  selected  points  by  recording  gages  operated  continuously 
day  and  night  for  many  years  by  the  United  States  Lake  Survey.  The  most 
important  gages  are  at  Tibbetts  Point  (New  York),  Buffalo  (New  York), 
Cleveland  (Ohio),  Harbor  Beach  (Michigan),  Mackinaw  City  (Michigan), 
Milwaukee  (Wisconsin),  and  Marquette  (Michigan).  As  the  investigation 
progressed,  it  gradually  became  more  clearly  evident  that  the  largest  and 
most  serious  errors  encountered  were  those  .which  arise  from  the  fact  that 
the  surface  of  any  one  of  the  Great  Lakes  at  any  given  instant  is  not  level 
except  by  accident.  The  surface  has  a  slope  at  every  point  due  to  the  in- 
fluence of  winds  and  barometric  pressures.  Hence,  when  the  elevation  of 
the  surface  of  Lake  Erie,  for  example,  is  observed  at  Buffalo  on  each  day,  it  is 
necessary,  if  one  is  to  obtain  daily  values  of  the  elevation  of  the  mean  surface 
of  Lake  Erie,  to  apply  a  correction  each  day  to  correct  for  the  disturbance  of 
the  elevation  on  that  day  at  Buffalo  produced  by  the  winds  and  barometric 
pressures. 

The  recording  gages  are  so  designed  and  operated  that  the  rapid  fluctua- 
tions of  elevation  of  the  water  surface  which  one  sees  as  wind  waves  are 
practically  eliminated  from  the  record.  The  elevation  recorded  by  the  gage 
at  any  instant  is  with  a  high  degree  of  accuracy,  usually  within  0.1  or  even 
within  0.01  foot,  equal  to  the  mean  elevation  for  the  preceding  10  minutes, 
even  though  the  wind  waves  may  be  causing  the  instantaneous  elevation  of 
the  water  at  the  gage  to  vary  through  a  range  of  several  feet  several  times  in 


BAROMETRIC   PRESSURES   ON   THE   GREAT  LAKES  3 

«ach  minute.  The  influence  of  the  slopes  and  differences  of  elevation  visible 
to  the  eye  as  waves  are  eliminated  automatically,  and  with  an  exceedingly 
high  degree  of  accuracy,  by  the  gages.  Such  slopes  are  local  in  character, 
they  extend  continuously  without  change  of  sign  for  a  few  feet  only,  and  at 
any  one  spot  on  the  lake  surface  they  persist  of  one  sign  for  a  few  seconds 
only.  The  mean  elevation  of  the  water  surface  as  recorded  by  a  gage  is  not 
-appreciably  affected  by  these  rapid  fluctuations  of  slope  and  elevation. 

Omit  from  consideration  the  rapid  and  large  fluctuations  of  slope  and 
elevation  of  short  periods  referred  to  in  the  preceding  paragraph.  Just  as 
the  effects  of  these  visible  wind  waves  are  automatically  eliminated,  by  the 
gages,  from  the  record  written  by  them,  so  imagine  the  surface  of  the  lake  as 
a  whole  to  be  freed  from  the  visible  wind  waves,  and  imagine  each  part  of  the 
surface  to  be  at  its  mean  elevation — a  mean  covering  say  10  to  15  minutes. 
This  smoothed-out  lake  surface  on  any  one  of  the  Great  Lakes  will  not  at 
any  time  be  level  except  by  accident.  It  will  always  be  disturbed  by  wind 
and  by  barometric  pressures.  But  the  slope  in  that  surface  will  always  and 
«verywhere  be  too  small  to  be  visible  to  the  unaided  eye.  Anywhere  on  the 
Great  Lakes  where  the  depth  of  water  is  as  much  as  10  feet  and  under  the 
most  extreme  conditions  of  wind  and  barometric  pressure  such  slopes  are 
probably  not  greater  than  0.011  foot  vertical  to  one  mile  horizontal.  The 
slopes  are  usually  less  than  0.002  foot  vertical  to  one  mile  horizontal  wherever 
the  depth  of  water  is  20  feet  or  more;  they  are,  however,  as  a  rule  continuous 
with  one  sign  over  the  whole  surface  of  the  lake  at  any  instant,  though  at 
times  there  is  one  reversal  of  sign  in  passing  from  one  end  (or  side)  of  a  lake 
to  the  other.  That  is,  in  general,  if  the  slope  produced  by  the  wind  and 
barometric  pressures  combined  is  downward  from  Buffalo  to  the  westward  at 
a  given  instant,  it  is  downward  to  the  westward  continuously  all  the  way  to 
the  west  end  of  the  lake.  Occasionally,  and  for  short  periods  only,  the  slope 
of  the  character  referred  to  may  be  downward  to  the  westward  in  a  part  of 
Lake  Erie  and  upward  to  the  westward  in  the  remaining  part,  or  vice  versa. 
The  slopes  of  the  character  referred  to  in  this  paragraph  persist  with  one 
sign  for  many  hours,  as  a  rule,  and  sometimes  for  several  days,  until  the  wind 
changes  or  the  barometric  conditions  change. 

It  is  these  long-continuous,  invisible  slopes  of  the  water  surface,  produced 
by  winds  and  differences  of  barometric  pressures,  that  cause  the  recorded 
mean  elevation,  for  any  hour  or  any  day,  at  a  gage,  to  be  different  from  the 
actual  mean  elevation  of  the  whole  surface  of  the  lake  for  that  hour  or  that 
day.  The  difference  between  the  mean  elevation  of  the  water  surface  at  a 
gage  for  any  hour  or  any  day  and  the  mean  elevation  of  the  whole  lake  sur- 
face during  that  same  time  is  the  combined  wind  effect  and  barometric 
effect.  This  publication  deals  with  the  investigation  of  these  effects  of  winds 
and  of  barometric  pressures.  It  shows  how  the  investigation  has  been  made 
and  how  corrections  may  be  applied  to  eliminate  the  major  part  of  the 
effects  of  winds  and  of  barometric  pressures,  and  so  to  secure  much  more 
accurate  values  than  are  otherwise  obtainable  for  the  mean  elevation  of 


4  EFFECTS   OF  WINDS   AND    OF 

the  whole  surface  of  each  lake.  The  manner  in  which  the  conclusions 
reached  in  this  investigation  may  be  applied  to  other  lakes  or  bodies  of  water 
and  may  help  in  solving  various  problems  is  briefly  indicated. 

DATA  USED  AND  ACKNOWLEDGMENTS. 

No  new  observations  were  made  in  this  investigation.  The  necessary 
observations,  in  great  abundance,  and  of  adequate  accuracy  and  reliability, 
had  already  been  made.  The  principal  data  used  include  hourly  and  daily 
observed  elevations  of  the  surface  of  Lake  Erie  and  Lake  Michigan-Huron 
at  five  gage  stations,  observed  hourly  wind  directions  and  velocities  at  five 
points  near  these  two  lakes,  and  the  observed  barometric  pressures  twice 
per  day  at  six  points. 

The  hourly  elevations  of  the  water  surface  which  were  used  on  Lake  Erie 
covered  48  selected  days  at  Buffalo  and  52  selected  days  at  Cleveland.  On 
Lake  Michigan-Huron  the  hourly  elevations  of  the  water  surface  which  were 
used  covered  34  days  at  Milwaukee,  42  days  at  Harbor  Beach,  and  42  days 
at  Mackinaw. 

The  daily  mean  elevations  of  the  water  surface  which  were  used  on  Lake 
Erie  covered  the  months  August-October  1909  and  June-October  1910 — 
eight  months  in  all.  Each  daily  mean  was  the  mean  of  24  hourly  values  for 
that  day  obtained  from  the  records  of  the  automatic  gages  which  had 
operated  at  Buffalo  and  Cleveland.  The  two  sets  of  daily  means  were 
entirely  separate  and  independent  for  the  two  stations  Buffalo  and 
Cleveland. 

On  Lake  Michigan-Huron  the  daily  mean  elevations  of  the  water  surface 
which  were  used  were  separate  and  independent  series  at  each  of  the  three 
stations,  Milwaukee,  Mackinaw,  and  Harbor  Beach,  each  covering  the 
months  June-September  1910  and  June-September  1911 — eight  months  in 
all.  Each  daily  mean  was  the  mean  of  24  hourly  values  for  that  day  obtained 
from  an  automatic  gage. 

The  observed  hourly  winds  which  were  used  had  been  obtained  from 
recording  anemometers  and  wind  vanes  as  operated  at  the  U.  S.  Weather 
Bureau  stations  at  Buffalo,  Cleveland,  Milwaukee,  Port  Huron  (Michigan) 
and  Sault  Ste.  Marie  (Michigan).  For  each  hour  the  total  travel  of  wind  in 
that  hour  was  observed  and  the  prevailing  direction  to  the  nearest  of  the 
principal  points,  N,  NE,  E,  SE,  etc.,  at  intervals  of  45°.  The  hourly  winds 
used  at  the  stations  named  cover  all  of  the  months  named  in  connection 
with  the  daily  mean  elevations  of  water  surface  and  also  all  of  the  separate 
days  for  which  hourly  water  elevations  were  used. 

The  observed  barometric  pressures  at  six  stations  near  the  two  lakes  were 
read  from  the  isobars  (lines  of  equal  barometric  pressure)  as  shown  on  the 
daily  forecast  maps  as  prepared  at  the  U.  S.  Weather  Bureau  at  Chicago  for 
the  use  of  the  forecaster  at  that  station.  Two  such  maps  are  made  for  each 
day,  one  showing  the  facts  at  8  a.m.,  75th  meridian  time,  and  the  other  facts 
at  8  p.m.,  75th  meridian  time.  The  barometric  pressures  used  in  this  investi- 
gation were  for  the  two  times  named  on  each  of  the  dates  covered  by  the 


BAROMETRIC  PRESSURES  ON  THE  GREAT  LAKES       5 

daily  mean  elevations  of  the  water  surface  and  by  the  hourly  elevations  of 
the  water  surface  which  have  been  mentioned. 

The  above  statement  shows  what  data  were  fully  used  in  the  portion  of  the 
investigation  covered  specifically  by  this  publication.  Many  more  data  of 
these  kinds  have  been  collected  and  used  to  furnish  general  checks  on  the 
conclusions  reached.  For  example,  many  more  daily  mean  elevations  of 
water  surface  at  various  stations  were  collected  and  examined.  So,  too,  the 
barometric  pressures  have  been  secured  and  examined  at  eleven  stations  in 
the  vicinity  of  the  Great  Lakes  for  the  two  times  named  above  on  each  day 
in  28  months  during  the  years  1909  to  1913,  inclusive. 

Grateful  acknowledgment  is  hereby  extended  to  the  two  organizations 
which  have  furnished  the  data  used  in  this  investigation  and  to  several  per- 
sons in  those  organizations  who  have  been  especially  helpful  in  supplying 
information  and  valuable  suggestions. 

The  United  States  Weather  Bureau  has  furnished  all  meteorological  data 
promptly  or  has  given  the  opportunity  to  get  it  from  the  original  records. 
The  Chief  of  the  Weather  Bureau,  C.  F.  Marvin,  has  taken  a  continuous 
interest  in  the  investigation,  has  supplied  many  comments  and  suggestions, 
and  has  given  freely  the  benefit  of  his  good  judgment.  Prof.  Henry  J.  Cox, 
meteorologist,  in  charge  of  the  Weather  Bureau  station  at  Chicago,  has 
freely  supplied  working  facilities  during  many  days  at  various  times  for  those 
who  have  been  working  on  this  investigation.  He  and  his  assistants  have 
facilitated  the  work  in  many  ways,  but  especially  by  granting  free  access  to 
the  forecast  maps  and  other  records  at  the  Chicago  office. 

The  United  States  Lake  Survey  (Survey  of  the  Northern  and  Northwest- 
ern Lakes)  has  furnished  freely  a  large  amount  of  information  in  regard  to 
hourly  and  mean  daily  elevations  of  the  water  surface  at  various  gage  sta- 
tions on  the  Great  Lakes,  of  which  that  used  in  the  particular  investigation 
treated  in  this  publication  is  but  a  part. 

Acknowledgment  is  hereby  made  to  the  various  officers  of  the  Corps  of 
Engineers  of  the  United  States  Army  who  have  been  in  charge  of  the  Lake 
Survey  at  the  various  times  when  data  have  been  requested  and  furnished. 
Especial  acknowledgment  is  also  extended  to  Mr.  F.  G.  Ray  and  to  Mr. 
Thomas  Russell,  of  the  Lake  Survey,  for  information  given  and  courtesies 
extended  at  various  times  in  connection  with  this  investigation.  Mr. 
Walter  J.  Graves,  formerly  of  the  Lake  Survey,  also  furnished  especially 
valuable  information  and  suggestions  early  in  the  investigation. 

Mr.  J.  A.  Folse  worked  on  this  investigation  as  a  computer  for  4,000  hours 
in  all,  intermittently,  with  many  interruptions  in  1913-20,  and  continuously 
from  June  1920  to  July  1921,  inclusive.  In  1920  and  1921  all  phases  of  the 
investigation  were  discussed  fully  with  him,  and  he  has  incidentally  given 
many  valuable  suggestions.  The  extensive  use  of  plotted  data,  studied  in 
the  graphic  form,  to  supplement  the  analyses  made  by  the  least-square 
method  of  computation,  has  been  due  largely  to  his  insistent  suggestions. 
Such  use  has  proved  to  be  very  effective  in  leading  to  a  true  understanding 
of  certain  phases  of  the  investigation. 


6  EFFECTS   OF   WINDS   AND   OF 

METHODS  OF  THIS  INVESTIGATION. 

In  general,  both  in  this  investigation  of  wind  effects  and  barometric  effects 
and  in  the  larger  investigation  of  evaporation  from  the  Great  Lakes,  of  which 
it  is  a  part,  the  procedure  has  been  as  indicated  in  the  following  numbered 
paragraphs : 

(1)  The  immediate  problem  to  be  attacked  was  studied  in  the  light  of  all 
available  information.     A  theory  was  developed  as  to  the  relations  of  the 
various  quantities  involved.     The  theory  was  expressed  in  the  form  of  ,a 
general  equation.     The  equations  were  then  set  up  for  a  least-square  solu- 
tion to  test  that  theory.     Each  equation  expressed  an  observed  quantity, 
a  change  in  elevation  of  the  water  surface,  in  terms  of  other  known  or  ob- 
served quantities,  in  conformity  with  the  theory  which  was  on  trial.     The 
solution  was  then  made  by  the  least-square  method  of  computation.     The 
principal  outcome  was  (a)  a  set  of  computed  values  of  the  unknowns, 
assumed  to  be  constants,  which  were  supposed  to  express  the  relations 
between  the  observed  quantities,  and  (6)  a  set  of  residuals  which  are  the 
discrepancies  between  the  tentative  theory  and  the  observed  facts. 

(2)  The  outcome  of  the  least-square  solution  was  then  studied  in  the  light 
of  all  available  internal  and  external  evidence.  As  the  theory  expressed  in  the 
observation  equations  of  such  a  solution  approaches  more  closely  to  perfec- 
tion and  to  completeness,  the  computed  probable  errors  are  smaller,  the 
residuals  as  a  rule  are  smaller,  and  the  distribution  of  the  residuals  as  to  sign 
and  magnitude  follow  the  laws  of  accidental  errors  more  closely.     These 
tests  furnished  the  main  portion  of  the  internal  evidence.     The  external 
evidence  was  derived  mainly  (a)  from  comparisons  of  the  outcome  of  the 
solution  with  that  from  other  solutions  already  made,  (6)  from  a  study  of 
apparently  abnormal  residuals,  and  (c)  from  general  checks  on  the  reliability 
of  the  various  items  of  outcome  of  the  solution  which  were  derived  from 
any  available  information   which   was  independent   of  the   least-square 
solutions. 

(3)  In  some  cases,  especially  in  the  later  portions  of  the  investigation,  the 
observed  fluctuations  in  elevation  of  the  water  surface  were  plotted  in  graphs, 
and  the  fluctuations  as  computed  from  the  constants  derived  from  the  least- 
square  solutions  were  plotted,  superposed,  on  the  same  scale.     The  graphs 
were  then  studied  to  secure  further  checks,  contradictions,  or  suggestions. 

(4)  In  the  light  of  all  evidence  a  new  set  of  observation  equations  was  then 
set  up  and  the  whole  process  repeated  of  making  a  least-square  solution  and 
then  studying  the  outcome  from  it.     In  general,  each  new  set  of  observation 
equations  involved  one  or  more  of  the  following:  (a)  a  change  in  the  tentative 
theory,  expressed  as  a  change  in  the  form  of  the  observation  equations;  (6) 
a  change  in  the  data  used,  brought  about  by  rejecting  certain  observation 
equations  or  by  combining  certain  others  (two  or  more  in  a  group)  to  form 
one;  (c)  the  addition  of  a  considerable  amount  of  data,  so  as  to  increase  the 
number  of  observation  equations  to  at  least  double  what  it  had  been,  or  (d) 


BAROMETRIC   PRESSURES   ON   THE    GREAT   LAKES  7 

the  new  observation  equations  were  based  upon  an  entirely  independent 
group  of  data,  elevations  of  water  surface,  from  a  different  gage,  which  might 
even  be  on  a  different  lake. 

Though  the  general  procedure  was  that  indicated  above,  the  arrangements 
were  such,  especially  when  several  computers  were  working  at  the  same  time, 
that  two  or  more  least-square  computations  were  in  progress  simultaneously, 
relating  to  different  gages  or  to  different  phases  of  the  investigation.  In  gen- 
eral, the  investigation  has  been  so  managed  that  each  general  conclusion 
adopted  depends  upon  two  or  more  least-square  solutions  and  the  corre- 
sponding studies  based  upon  independent  sets  of  data,  usually  from  different 
gages  and  if  feasible  from  different  lakes. 

The  least-square  method  of  solution  was  adopted  as  the  principal  method 
of  attack  on  this  problem  because  it  was  apparent  that  several  different  fac- 
tors or  influences  were  operating  simultaneously  to  cause  fluctuations  of  the 
elevation  of  the  water  surface  at  a  gage,  no  one  of  which  could  be  safely 
neglected  while  attempting  to  evaluate  others.  In  this  investigation,  deal- 
ing with  the  facts  on  the  full  scale  of  nature  and  under  uncontrolled  natural 
conditions,  it  is  not  feasible  to  use  the  familiar  laboratory  strategy  of  elimi- 
nating from  the  phenomena  by  control  the  influence  of  all  factors  except  one, 
determining  the  influence  of  that  one,  and  then  doing  likewise  for  each  of 
the  other  factors  in  turn.  Hence,  it  was  believed  that  the  best  method  of 
attack  in  this  case  is  the  least-square  method  of  computation.  That  method 
is  especially  adapted  to  taking  into  account  simultaneously  several  control- 
ling factors  and  to  determining  simultaneously  their  separate  influences. 
The  outcome  has  fully  justified  this  method. 

In  the  broader  investigation  of  evaporation,  including  the  investigation 
of  wind  effects  and  barometric  effects  which  is  here  reported,  74  complete 
least-square  solutions  and  the  corresponding  studies  have  been  made. 
Typical  solutions  each  contained  from  100  to  600  observation  equations, 
each  having  for  its  absolute  term  the  observed  change  in  elevation  of  the 
water  surface  at  a  gage  during  a  day  or  an  hour,  and  each  containing  from  2 
to  8  unknown  constants  to  be  determined.  In  an  extreme  solution  each 
observation  equation  contained  40  unknowns,  and  there  were  619  observa- 
tion equations  in  the  set. 

Of  the  74  least-square  solutions  referred  to  in  the  preceding  paragraph,  9 
were  directly  utilized  in  determining  the  wind  effects  and  barometric  effects. 
All  of  the  remainder  contributed  more  or  less  indirectly  by  gradually 
leading  the  investigator  toward  a  true  understanding  of  these  effects  and 
therefore  toward  a  true  expression  of  the  theory. 

The  statement  just  made,  indicating  the  large  number  of  least-square 
solutions  made  and  their  complexity,  shows  this  to  have  been  an  extensive 
investigation.  More  than  22,500  man-hours  of  time  have  been  spent  on 
the  routine  part  of  the  computations  and  studies  connected  with  the  broad 
investigation  of  evaporation,  including  the  investigation  of  wind  and  baro- 
metric effects.  From  1  to  10  persons  have  been  engaged  in  this  routine  work 


8  EFFECTS   OF   WINDS   AND   OF 

at  any  one  time  at  various  intervals  since  it  was  commenced  in  the  summer 
of  1911.  In  all,  31  persons  have  taken  part.  I  have  spent  more  than  2,000 
hours  on  the  investigation,  none  in  routine  work. 

OUTCOME  OF  THE  INVESTIGATION. 

The  outcome  of  the  investigation  may  be  very  briefly  characterized  as 
follows: 

(1)  Reasonably  accurate  numerical  expressions  have  been  obtained  for 
the  effects  of  barometric  pressures  on  the  elevation  of  the  water  surface  at 
the  five  stations,  Buffalo,  Cleveland,  Milwaukee,  Mackinaw,  and  Harbor 
Beach,  on  Lake  Erie  and  on  Lake  Michigan-Huron.    With  these  expressions, 
one  may,  from  the  distribution  of  barometric  pressures  ordinarily  shown  on 
the  forecast  maps  of  the  Weather  Bureau,  compute  the  disturbances  in 
elevation  of  the  water  surface  thereby  produced  at  the  stations  named. 

(2)  The  general  method  has  been  developed  by  which  such  a  numerical 
expression  for  the  barometric  effect  at  any  station  on  any  body  of  water  may 
be  derived  from  observations  of  the  water  elevation  at  that  station  and  the 
forecast  maps  for  the  same  period. 

(3)  A  general  expression,  including  the  necessary  numerical  constant,  has 
been  obtained  for  the  effect  of  winds,  of  any  given  velocity  and  direction,  in 
producing  a  disturbance  of  elevation  of  the  water  surface  at  any  given  sta- 
tion, on  any  body  of  water,  anywhere  in  the  world.     The  data  required  in 
regard  to  the  station  and  the  body  of  water  are  such  as  are  ordinarily  shown 
on  good  charts,  namely,  the  depths  of  the  water  at  all  points,  the  location  of 
the  shore  line,  and  the  location  of  the  station. 

(4)  Four  of  the  prevailing  seiches,  or  free  oscillations  under  the  influence 
of  inertia,  on  Lake  Erie  and  Lake  Michigan-Huron  have  been  isolated. 
Their  periods  and  probable  methods  of  oscillation  have  been  shown.     The 
relation  between  these  seiches  and  the  uncertainties  in  daily  mean  elevations 
of  the  water  surface  at  gage  stations  has  been  discerned.     The  appreciation 
of  this  relation  aids  decidedly  in  obtaining  accurate  determinations  of  the 
daily  mean  elevation  of  the  mean  surface  of  each  lake. 

(5)  The  accuracy  with  which  the  elevation  of  the  mean  surface  of  any  one 
of  the  Great  Lakes  may  be  determined  for  any  given  day  has  been  decidedly 
increased.     On  Lake  Erie  the  elevation  of  the  mean  surface  of  the  lake  may 
now  be  determined  as  accurately  from  one  day  of  observation  at  Buffalo 
as  it  was  formerly  possible  to  fix  it  from  16  days  of  observation  at  that  sta- 
tion.    Similarly,  the  elevation  of  the  mean  surface  of  Lake  Michigan-Huron 
may  now  be  determined  as  accurately  from  one  day  of  observation  at  Mack- 
inaw as  it  was  formerly  possible  to  determine  it  from  6  days  of  observation 
at  that  station.     When  one  determines  the  fluctuation  of  elevation  of  the 
mean  surface  of  a  lake  he  thereby  determines  the  fluctuation  in  the  total 
water-content  of  the  lake. 

(6)  The  relations  of  the  new  knowledge  indicated  in  (1)  to  (5)  to  four  out- 
standing problems  have  become  evident.    The  four  problems  are: 


BAROMETRIC    PRESSURES    ON   THE    GREAT   LAKES  9 

(a)  The  problem  of  regulating  the  elevations  of  the  water  surface  of  each 
of  the  Great  Lakes — and  the  rates  of  flow  through  the  connecting  streams, 
so  as  to  secure  the  greatest  aggregate  benefits  to  navigation,  power,  develop- 
ment, and  sanitation. 

(6)  The  problem  of  determining  the  laws  of  evaporation  from  large  free- 
water  surfaces  such  as  the  surface  of  the  Great  Lakes. 

(c)  The  problem  of  correcting  the  observed  elevations  of  the  water- 
surface  at  a  tide-gage  in  such  a  manner  as  to  remove  the  disturbances  due 
to  winds  and  fluctuating  barometric  pressures  and  thereby  to  secure  a  more 
accurate  determination  of  mean  sea-level  than  could  otherwise  be  obtained 
from  said  observations. 

(d)  The  problem  of  determining  the  direction  and  rate  of  the  tilting, 
which  is  believed  to  be  in  progress,  of  the  land  underlying  and  immediately 
surrounding  the  Great  Lakes. 

ORDER  OF  PRESENTATION. 

The  order  of  presentation  in  this  publication  is  briefly  indicated  in  the 
following  paragraphs: 

(1)  The  manner  in  which  the  least-square  solutions  which  serve  to  deter- 
mine barometric  effects  were  set  up  is  first  given,  including  the  theoretical 
basis  of  the  observation  equations.     The  principal  facts  in  regard  to  the  five 
final  barometric  solutions  are  given,  including  the  values  computed  from 
them.     The  manner  of  using  these  values  to  compute  hourly  and  daily 
barometric  effects  is  set  forth. 

(2)  Similarly,  the  manner  of  setting  up  the  least-square  solutions  which 
serve  to  determine  the  wind  effects  is  given,  together  with  the  principal  facts 
in  regard  to  the  four  final  wind  solutions  and  the  values  computed  from  them. 
The  formula,  constants,  and  method  for  computing  hourly  wind  effects  are 
set  forth.     The  method  used  in  computing  daily  wind  effects  is  shown. 

(3)  The  accuracy  of  the  computed  barometric  effects  is  discussed,  using 
both  the  internal  evidence  of  the  computations  and  external  evidence. 

(4)  Similarly,  the  accuracy  of  the  computed  wind  effects  is  discussed. 

(5)  The  evidence  as  to  the  over-all  accuracy  attained  in  the  attempt  to 
secure  elevations  of  the  mean  surface  of  a  whole  lake  by  applying  corrections 
for  wind  effects  and  barometric  effects  at  the  gage  stations  is  discussed. 

(6)  The  evidence  obtained  in  regard  to  seiches,  free  oscillations  under  the 
influence  of  inertia,  in  Lake  Erie  and  Lake  Michigan-Huron,  is  set  forth  and 
discussed. 

(7)  Certain  generalizations  are  made  as  to  wind  effects,   barometric 
effects,  and  seiches.     These  generalizations  are  intended  to  help  one,  during 
a  first  reconnaissance  of  a  problem  connected  with  any  lake  (or  other  large 
body  of  water),  to  form  a  first  approximate  estimate  as  to  the  probable 
magnitude  and   character   of   the   wind   effects,  barometric   effects,  and 
seiches  on  that  lake  and  their  probable  bearing  on  the  problem  to  be  at- 
tacked. 


10  EFFECTS   OF   WINDS   AND   OF 

(8)  The  relation  of  the  research  to  four  important  outstanding  problems 
in  science  and  engineering  is  briefly  indicated. 

The  table  of  contents  at  the  beginning  of  this  publication  is  intended  to 
give  a  good  general  view  of  the  order  of  presentation  in  more  detail  than  the 
preceding  statement;  it  will  assist  especially  the  reader  who  is  referring 
back  to  the  publication  to  look  up  a  particular  topic. 

THEORETICAL   BASIS    FOR    BAROMETRIC-OBSERVATION 

EQUATIONS. 

What  is  the  shape  of  the  surface  of  a  lake  when  its  water  is  in  equilibrium 
under  the  influence  of  gravity  and  barometric  pressures?  The  wind  is  ig- 
nored in  this  question.  The  answer  is  evidently  the  desired  fundamental 
theoretical  basis  for  a  study  of  barometric  effects — disturbances  of  elevation 
of  the  water  surface  at  a  gage  station  produced  by  barometric  pressures. 

If  there  is  equilibrium  at  every  part  of  the  lake,  under  the  conditions  stated, 
with  no  wind  blowing,  according  to  the  fundamental  principle  of  hydrostatics 
the  pressure  at  every  point  in  the  lake  at  a  given  elevation  must  be  the  same 
as  at  every  other  point  at  that  elevation.  Also  that  pressure  must  be  at 
every  point,  x, 

p  =  (H,-HJ8w+M8m  (1) 

in  which 

Hs  is  the  elevation  of  that  part  of  the  surface  of  the  water  which  is 

directly  above  the  point. 
Hx  is  the  elevation  of  the  point. 
dw  is  the  density  of  the  water. 
M  is  the  length  of  the  mercury  column  which  measures  the  barometric 

pressure  upon  the  surface  of  the  water  above  the  point. 
Sm  is  the  density  of  mercury. 

Ht—Hx  is  the  distance  from  the  surface  of  the  water  down  to  point  X. 
(Ht—Hx)8w  is  the  pressure  at  the  point  X  due  to  the  weight  of  the  water 

above  it. 

Mdm  is  the  pressure  at  the  point  X  due  to  barometric  pressure  at  the 
surface  above,  which  pressure  is  necessarily  transmitted  to 
the  point  X  under  the  conditions  stated. 

Note  that  M  is  the  ordinary  expression,  incorrectly  used,  for  the  baromet- 
ric pressure,  namely,  the  length  of  mercury  column  which  will  balance  the 
barometric  pressure,  say  30  inches,  under  certain  conditions. 

Consider  the  relations  between  the  quantities  Hs  and  M  for  two  points,  1 
and  2,  at  the  same  elevation  Hx  in  the  water  of  the  lake,  under  the  conditions 
of  equilibrium  which  have  been  stated.  Let  the  elevation  of  the  water  sur- 
face be  called  HI  at  point  1  and  H^  at  point  2.  Let  the  pressures  at  the  two 
points  be  called  pi  and  p2,  respectively.  Let  the  barometric  pressures  as 
ordinarily  expressed,  in  terms  of  M ,  at  the  water  surface  above  the  two  points 
be  called  M i  and  Mz,  respectively. 


BAROMETRIC   PRESSURES   ON   THE    GREAT  LAKES  11 

For  the  condition  of  equilibrium  pi  =  p2  and  therefore  from  equation 
(1)  it  is  clear  that 

(Ht-H^+Mj^  (Ht-HJl.+Mj*  (2) 

From  equation  (2)  it  follows,  by  cancellation  and  rearrangement  of  terms, 

that  #!  -  tf  2  =  -  (Mi  -  M2)  ^  (3) 

Qw 

Equation  (3)  expresses  the  fact  that  under  conditions  of  equilibrium  the 
contour  lines  (lines  of  equal  elevation)  on  the  surface  of  the  water  must 
coincide  in  shape  with  the  isobars  (lines  of  equal  barometric  pressure  at  the 
surface  of  the  water),  that  increasing  elevations  from  contour  to  contour 
must  correspond  to  decreasing  barometric  pressures  from  isobar  to  isobar, 
and  that  a  unit  interval  between  contours  must  correspond  to  an  interval  of 

—  between  isobars. 
Sw 

The  elevation  of  the  water  surface  on  the  Great  Lakes  is  ordinarily  ex- 
pressed in  feet  above  mean  sea-level.  HI — H»  is  therefore  most  conveniently 
expressed  in  feet.  The  barometric  pressures  are  usually  expressed  by  the 
U.  S.  Weather  Bureau  in  inches  of  mercury  at  0°C.,  and  were  so  recorded  on 
the  forecast  maps  used  in  this  investigation.  The  density  of  mercury  at 
0°C.  =  13.6.  The  water  concerned  in  equation  (3)  is  the  surface  water  of 
the  lakes,  of  which  the  temperature  will  seldom  be  outside  the  limits  32°F.  = 
0°C.  and  80°F.  =  27°C.  Therefore  the  density  of  this  water  will  seldom  be 
outside  the  extreme  limits  1.000,  at  39°F.  =4°C.  and  0.997,  at  80°F.  =27°C. 
With  sufficient  accuracy  for  the  present  purpose  the  density  of  the  water 

may  be  assumed  to  be  constant  at  1.00,  and  —  may  therefore  be  assumed 

8W 

to  be  constant  at  13.6. 

For  convenience,  recognizing  the  units  ordinarily  used  for  elevations  and 
for  barometric  pressures,  equation  (3)  may  now  be  rewritten  thus 

H.-H^  -(M1-MJ)(13.6)(TJj)  =  -(M1-Af2)(1.13)  (4) 

Note  that  the  division  by  12  in  the  second  member  of  the  equation  is  to 
take  into  account  the  fact  that  the  M's  are  expressed  in  inches  whereas  the 
H's  are  expressed  in  feet. 

The  upper  part  of  plate  1  shows  the  isobars  over  Lake  Erie  and  vicinity  as 
obtained  from  the  forecast  map  of  the  U.  S.  Weather  Bureau  as  used  at  the 
Chicago  office  of  the  Bureau  for  8  p.m.  on  August  5,  1910.  Note  that  the 
interval  between  isobars  is  0.01  inch,  that  the  isobars  are  nearly  straight 
across  Lake  Erie,  and  that  the  barometric  gradient  is  downward  to  the 
northeastward  over  Lake  Erie. 

The  middle  part  of  plate  1  shows  the  contours  of  the  surface  of  the  water  on 
Lake  Erie  at  8  p.m.  on  August  5,  1910,  in  accordance  with  equation  (4). 
These  contours  would  have  existed  if  no  wind  had  been  blowing  at  that  time 
and  if  the  water  of  Lake  Erie  had  been  in  equilibrium  at  that  time  under  the 
influence  of  gravity  and  barometric  pressure.  Note  that  an  arbitrary  zero 


12  EFFECTS   OF   WINDS   AND    OF 

has  been  adopted  for  these  contours  coinciding  with  the  location  of  the  isobar 
marked  29.90.  Note,  also,  that  the  interval  between  contours  is  0.0113 
foot,  which  corresponds,  in  accordance  with  equation  (4),  to  the  interval  of 
0.01  inch  between  the  isobars  shown  on  the  upper  part  of  the  plate.  Note, 
that  the  surface  gradient  of  the  water  of  Lake  Erie  is  shown  as  upward  to  the 
northeastward  in  accordance  with  the  barometric  gradient  shown  on  the 
upper  part  of  the  plate  as  downward  to  the  northeastward. 

Plate  1  is  a  concrete  illustration  of  the  relation  between  isobars  and  surface 
contours  at  any  water  surface  which  must  exist  if  the  water  is  in  equilibrium 
under  the  influence  of  gravity  and  barometric  pressure. 

BAROMETRIC  EFFECTS  IN  TERMS  OF  BAROMETRIC  PRESSURES. 

In  setting  up  observation  equations  to  express  the  relation  between  ob- 
served fluctuations  in  water  elevation  and  barometric  pressures  as  shown  on 
the  forecast  maps  it  would  be  desirable  to  utilize  accurately  the  relations 
shown  in  equation  (4)  and  illustrated  in  plate  1,  if  limits  of  time,  expense, 
and  accuracy  did  not  prevent  it  from  being  feasible.  But  a  reconnaissance 
of  the  problem  (including  an  attempt,  on  a  small  scale,  to  utilize  the  exact 
relations)  indicated  that  in  order  to  keep  within  limits  imposed  on  this  in- 
vestigation it  was  necessary  to  introduce  an  assumption,  called  assumption 
No.  1. 

ASSUMPTION  No.  1. 

It  is  assumed  that,  with  sufficient  accuracy  for  the  purposes  of  this  in- 
vestigation, the  isobars  at  the  instant  represented  by  any  forecast  map 
(8  a.m.  or  8  p.m.,  75th  meridian  time)  are  straight  and  uniformly  spaced 
within  the  limits  of  the  lake  under  consideration. 

Assumption  No.  1  is  nearly  true  for  the  actual  case  illustrated  by  plate  1. 
In  general,  for  each  of  the  Great  Lakes  assumption  No.  1  is  nearly  true. 
The  size  of  any  one  barometric  low-pressure  area  or  of  any  high-pressure 
area  is  usually  many  times  as  great  as  that  of  any  one  of  the  lakes.  Within 
the  area  occupied  by  the  lake  the  isobars  curve  but  little  and  there  is  but 
moderate  departure  from  uniform  spacing.  The  most  serious  departures 
from  assumption  No.  1  in  these  two  respects  ordinarily  occur  when  the  center 
of  a  well-developed  low-pressure  area  is  over  a  lake.  One  may  verify  these 
statements  by  studying  the  forecast  maps. 

Assumption  No.  1  was  introduced  to  save  time  and  expense.  The  ulti- 
mate effect  of  its  introduction  in  reducing  the  accuracy  of  the  final  computed 
barometric  effects  is  believed  to  be  moderate  only. 

Assumption  No.  1  combined  with  the  relations  between  isobars  and  con- 
tours on  the  water  surface  which  have  been  commented  upon  and  which  are 
fixed  by  equation  (4)  makes  the  contours  on  the  water  surface  straight  and 
uniformly  spaced.  In  other  words,  under  assumption  No.  1,  whenever  the 
water  is  in  equilibrium  under  the  influence  of  gravity  and  barometric  pres- 
sures its  surface  is  plane. 

In  the  case  illustrated  in  plate  1,  for  the  actual  isobars  there  shown  one 


BAROMETRIC   PRESSURES   ON   THE   GREAT   LAKES  13 

must  now,  under  assumption  No.  1,  substitute  other  isobars  coinciding  as 
nearly  with  the  actual  isobars  as  is  consistent  with  the  condition  that  they 
must  everywhere  over  Lake  Erie  be  straight  and  uniformly  spaced.  The 
corresponding  contours  on  the  lake  surface,  for  the  conditions  of  equilibrium, 
will  be  straight  and  uniformly  spaced  as  indicated  in  the  lower  part  of  plate  1. 

The  total  amount  of  water  in  Lake  Erie  is  not  affected  by  the  barometric 
pressure,  or  its  distribution.  The  effect  of  the  greater  barometric  pressure 
on  the  southwestern  part  of  the  lake  as  compared  with  that  on  the  north- 
eastern part  of  the  lake  is  to  subtract  water  from  the  southwestern  part  and 
add  it  to  the  northeastern  part.  Some  line  on  the  surface  of  the  water,  such 
as  that  marked  as  the  nodal  line  on  the  lower  part  of  Plate  1,  is  not  changed 
in  elevation.  What  is  the  location  of  that  line?  The  direction  of  the  nodal 
line  is  evidently  parallel  to  the  contours.  It  remains  to  fix  one  point  on  the 
nodal  line. 

Consider  an  elementary  portion  of  the  lake  surface  as  shown  on  the  lower 
sketch  on  plate  1  of  which  the  two  dimensions  are  8L  at  right  angles  to  the 
contours  and  8W  parallel  to  the  contours,  of  which  the  area  is  8L8W  =  8A 
and  of  which  the  distance  from  the  nodal  line  is  L.  Let  the  slope  of  the 
water  surface  be  called  S. 

The  volume  of  water  which  had  been  added  at  this  area  by  the  barometric 
influence  is  Depth  of  added  water  (area)  =SL8L8W  =  SL8A 

SL  is  the  depth  of  water  added  at  the  particular  area. 
The  total  amount  of  water  added  to  the  northeastward  of  the  nodal  line 
is  the  integral  over  that  portion  of  lake  of  these  elementary  volumes,  namely: 

fsLdL8W  =  fsL8A  =  S  f L8A  (5) 

S  may  be  placed  outside  the  integral  sign,  as  it  is  the  slope  which  has  been 
assumed  to  be  constant. 

Consider  the  portion  of  the  lake  which  lies  to  the  southwestward  of  the 
nodal  line,  from  every  portion  of  which  water  is  subtracted  by  the  barometric 
influence.  Use  the  same  notation  as  before,  but  let  L  be  counted  as  negative 
when  measured  to  the  southwestward  from  the  nodal  line.  Then  for  this 
portion  of  the  lake  the  total  amount  of  water  subtracted  is  the  same  integral 

as  before,  namely,  f 

S  I  L8A  (6) 


in  which,  however,  all  values  of  L  are  negative  and  the  integral  is  negative. 

As  the  total  amount  of  water  in  the  lake  has  not  been  changed,  the  sum  of 

integrals  (5)  and  (6)  must  be  zero — that  is,  the  amount  of  water  added  on  one 

side  of  the  nodal  line  must  equal  that  subtracted  from  the  other  side.     In 

other  words  the  integral  S  I  L8A  over  the  whole  of  the  lake  surface  must  be 
zero,  the  distance  L  being  reckoned  from  the  nodal  line  as  indicated.  Hence, 
S  being  a  constant,  /  L8A  over  the  whole  area  of  the  lake  must  be  zero. 


14  EFFECTS   OF   WINDS   AND   OF 

The  well-known  condition  which  locates  the  so-called  center  of  gravity  of 
an  area  is  that  the  integral  over  the  whole  area  /  LdA  is  zero  with  L  reck- 
oned from  any  line  through  the  center  of  gravity.  Hence,  it  is  clear  that 
the  nodal  line  under  assumption  No.  1  of  isobars  which  are  straight  and  uni- 
formly spaced  is  a  line  parallel  to  the  isobars  passing  through  the  center  of 
gravity  of  the  area  of  the  lake  surface. 

The  position  of  the  center  of  gravity  of  the  area  of  Lake  Erie  is  indicated 
in  the  lower  part  of  plate  1  by  the  circle  labeled  "  C.  G.  of  lake  area. "  It  is 
634,000  feet  west  and  276,000  feet  south  of  the  Buffalo  gage  station.  Its 
latitude  is  42°  07'  and  its  longitude  is  81°  13'. 

For  all  conditions  of  equilibrium  under  the  influence  of  gravity  and  baro- 
metric pressures,  under  the  restrictions  of  assumption  No.  1,  the  elevation  of 
the  lake  surface  at  the  center  of  gravity  remains  unchanged. 

Let  Hc  be  this  fixed  elevation  of  the  water  surface  at  center  of  gravity — 
that  is,  fixed  and  unchangeable  in  so  far  as  changes  of  barometric  pressure 
limited  by  assumption  No.  1  are  concerned.  Let  Mc  be  the  barometric 
pressure  on  the  surface  of  the  water  at  the  center  of  gravity.  Then  equation 
(4)  may  be  rewritten  thus: 

E1  =  H1-Hc=-(Ml-Mc)(l.l^  (7) 

in  which  E\  is  the  barometric  effect,  under  the  specific  conditions,  on  the 
elevation  of  the  water  surface  at  the  point  1. 

For  convenience  in  computation  it  is  now  proposed  to  express  Hi—Hc  in 
terms  of  slopes  of  the  water  surface  measured  along  parallels  and  meridians 
and  to  express  Mi—Mc  similarly  in  terms  of  barometric  gradients  measured 
along  parallels  and  meridians. 

The  barometric  gradient  between  two  points  is  the  difference  in  barometric 
pressures  at  the  two  points  divided  by  the  distance  between  the  points. 

Let  the  barometric  gradient  along  a  parallel  be  called  the  "W-E  gra- 
dient," and  let  it  be  called  positive  when  the  barometric  pressure  increases 
to  the  westward.  Similarly,  let  the  barometric  gradient  along  a  meridian 
be  called  the  "N-S  gradient,"  and  let  it  be  called  positive  when  the  baro- 
metric pressure  increases  to  the  northward. 

Let  the  co-ordinates  of  point  1  measured  from  the  center  of  gravity  of  the 
lake  along  parallels  and  meridians  be  Lw  along  a  parallel  and  Ln  along  a 
meridian  as  indicated  on  the  lower  part  of  plate  1.  Let  Lw  be  considered 
positive  to  the  eastward  and  Ln  positive  to  the  southward. 

Then,  keeping  in  mind  that  under  assumption  No.  1  the  isobars  are 
straight  and  equally  spaced, 

Ml-Mc=-(W-E  gradient)  CU)- (N-S  gradient) (LB)  (8) 

Similarly, 

#1  -  Hc  =  -  (W-E  slope)  (Li)  -  (N-S  slope)  (Ln)  (9) 

in  which  the  slope  of  the  water  surface  along  a  parallel  is  called  the  "W-E 
slope,"  positive  when  it  is  upward  to  the  westward,  and  the  slope  of  the 


BAROMETRIC   PRESSURES   ON   THE    GREAT   LAKES 


15 


water  surface  along  a  meridian  is  called  the  "N-S  slope,"  positive  when  it  is 
upward  to  the  northward.  Equation  (9)  is  true,  because  the  water  surface 
under  assumption  No.  1  is  an  inclined  plane. 

By  substitution  in  equation  (7)  of  the  values  of  Mi  —  Mc  and  of  H\—He 
from  equations  (8)  and  (9)  there  is  obtained 

Et  =  -  ( W-E  slope)  (Lw)  -  (N-S  slope)  (LB) 

=  +(W-E  gradient)(Lw)(1.13)  +  (N-S  gradient) (LB) (1.13)      (10) 

Or,  dropping  out  the  middle  part  of  (10),  which  is  no  longer  necessary  after 
the  conceptions  are  grasped 

#1  =  +  (W-E  gradient)  (Lw)  (1.13)+ (N-S  gradient) (LB)(  1.1 3)       (11) 

EXPRESSION  OF  BAROMETRIC  GRADIENTS. 

From  a  study  of  the  forecast  maps  and  of  the  general  conditions  of  the 
problem,  it  was  decided  that  the  best  feasible  way  to  secure  satisfactory 
values  for  the  W-E  gradient  and  the  N-S  gradient  for  each  of  the  Great 
Lakes  at  the  many  times  for  which  they  were  needed  was  as  follows : 

(1)  Eleven  points  were  selected   for   which 
readings  were  to  be  taken  from  the  forecast 
maps  to  cover  all  of  the  Great  Lakes  region. 
The  six  of  these  points  used  in  connection  with 
Lake  Erie  and  Lake  Michigan-Huron,  known 
as  points  3,  4,  5,  6,  7,  and  8,  are  shown  on  the 
sketch  on  the  lower  half  of  plate  2.     Their  exact 
locations  are  shown  in  the  adjoining  table: 

(2)  The  values  of  the  barometric   pressure 
were  read  directly  from  the  forecast  maps  for 

8  a.m.  and  8  p.m.  (75th  meridian  time)  of  each  day  and  tabulated  in  con- 
venient form. 

(3)  Let  "(6-8)"  stand  for  the  barometric  pressure  at  point  6  minus  the 
barometric  pressure  at  point  8  for  a  given  time.     Let  "  (distance  6  to  8) " 
stand  for  the  distance  from  point  6  to  point  8.     Note,  that  points  6  and  8 
are  on  the  same  parallel,  that  they  are  both  in  the  latitude  of  Lake  Erie, 
point  6  to  the  westward  and  point  8  to  the  eastward.     The  (W-E  gradient) 
for  Lake  Erie  was  then  taken  as 


Point. 

Lat. 

Long. 

3 

47|« 

85° 

4 

45 

87* 

5 

45 

80 

6 

42* 

85 

7 

40 

80 

8 

42* 

77* 

(6-8) 


(distance  6  to  8) 

(4)  Similarly,  the  (N-S  gradient)  for  Lake  Erie  was  taken  as 

(5-7) 


(distance  5  to  7) 
(5)  For  Lake  Michigan-Huron,  the  (W-E  gradient)  was  taken  as 

(4-5) 
(distance  4  to  5) 


(12) 


(13) 


(14) 


16  EFFECTS   OF   WINDS   AND    OF 

and  the  (N-S  gradient)  was  taken  as 

(3-6) 


(15) 


(distance  3  to  6) 

The  procedure  outlined  above  involves  assumption  No.  2,  which  is  stated 
in  the  following  paragraph: 

ASSUMPTION  No.  2. 

It  is  assumed  that  the  barometric  gradients  at  any  time  on  any  lake  along 
parallels  and  meridians  are  the  same  as  the  barometric  gradients  derived,  as 
indicated  above,  from  readings  taken  from  the  forecast  maps  at  the  selected 
points  3,  4,  5,  6,  7,  and  8,  which  lie  on  meridians  and  parallels  through  the 
lakes. 

It  is  believed  that  this  assumption  is  only  a  fairly  good  approximation, 
that  the  barometric  gradients  over  the  shorter  distances  limited  by  the  lake 
surface  vary  through  a  larger  range  than  the  gradients  over  the  longer  dis- 
tances between  the  reading  points  used  on  the  forecast  maps,  and  that  the 
variations  of  barometric  gradients  over  the  shorter  and  longer  distances  may 
not  be  in  step  —  that  is,  one  may  in  general  be  ahead  of  the  other.  It  is  be- 
lieved, however,  that  the  errors  in  assumption  No.  2  are  largely  canceled  out 
in  so  far  as  the  final  values  of  the  computed  barometric  effects  are  concerned. 
This  cancellation  is  believed  to  be  affected  in  part  by  the  device  of  introduc- 
ing the  proportionality  factors  Pw  and  Pn  into  the  derivation  of  the  observa- 
tion equations  as  indicated  later,  and  in  part  by  the  process  of  deducing,  from 
the  results  of  the  least-square  computations,  the  lag  of  barometric  effects 
behind  the  changes  in  barometric  gradients. 

For  Lake  Erie  the  expression  for  EI,  the  barometric  effect  at  any  point  1 
on  that  lake,  as  shown  in  equation  (11),  may  now  be  rewritten  as  follows  by 
means  of  expressions  (12)  and  (13): 


distance  6  to  8  /  \  distance  5  to  7 

=  +  (6-8)7^+(5-7)#n     (16) 
in  which 


H.-  *         and  R.  =        L13L»  (17) 

distance  6  to  8  distance  5  to  7 

Rw  and  Rn  are  constants  for  any  given  point  on  Lake  Erie.     They  differ  for 
different  points  but  do  not  change  with  the  lapse  of  time. 

PROPORTIONALITY  FACTORS  FOR  BAROMETRIC  EFFECTS. 
Equation  (16)  expressed  the  barometric  effect  on  the  elevation  of  the 
water  surface  at  any  given  point  1  on  Lake  Erie  provided  the  water  always 
remained  in  equilibrium  under  the  influence  of  gravity  and  barometric 
pressure.  The  general  conception  thus  far  set  forth,  and  including  assump- 
tion No.  1,  is  that  the  barometric  pressures  over  Lake  Erie  are  continually 


BAROMETRIC   PRESSURES   ON    THE   GREAT   LAKES  17 

changing  but  always  in  such  manner  that  the  isobars  over  the  lake  are 
straight  and  uniformly  spaced,  and  that  the  fluctuations  in  the  elevations  of 
different  parts  of  the  water  surface  take  place  at  once  in  such  a  manner  that 
the  water  is  always  in  equilibrium. 

It  is  obvious  that  the  effects  of  friction  and  of  inertia  will  tend  to  modify 
the  response  of  the  water  to  changing  barometric  pressures. 

Friction  will  tend  in  general  to  reduce  the  range  of  fluctuation  of  water 
surface  and  to  produce  a  lag  of  the  response  behind  the  barometric  changes 
which  produce  it. 

Inertia  will  tend  to  produce  an  initial  lag  in  the  response  of  the  water 
surface  to  any  change  in  the  barometric  pressure.  But  when  the  water  has 
once  started  from  one  part  of  the  lake  toward  another,  as  the  water  surface 
is  approaching  a  new  condition  of  equilibrium  after  some  relatively  sudden 
change  of  barometric  gradients,  inertia  will  tend  to  carry  the  water  past  the 
position  of  equilibrium  and  thereby  to  make  the  fluctuations  of  elevation  of 
the  water  surface  greater  than  those  corresponding  to  continuous  equilibrium. 

If  friction  is  relatively  large,  so  that  all  motions  of  the  water  produced  by 
inertia,  all  free  oscillations,  are  quickly  damped  out,  the  fluctuations  in  the 
elevation  of  the  water  surface  at  any  point  would  tend  to  be  considerably 
less  in  range  than  those  which  would  be  computed  from  equation  (16).  On 
the  other  hand,  if  friction  is  relatively  ineffective  in  damping  out  free  oscilla- 
tions of  the  water  of  the  lake  under  the  influence  of  inertia,  and  if  the  natural 
periods  of  oscillation  of  the  lake  happen  to  bear  certain  relations  to  the 
periods  of  change  in  the  barometric  gradients,  the  actual  fluctuations  in  the 
elevation  of  the  water  surface  at  a  point  might  largely  exceed  those  com- 
puted from  equation  (16). 

Hence,  aside  from  providing  later  for  an  assumed  lag  to  be  determined  by 
the  observations  themselves,  through  the  least-square  solution,  it  is  also 
advisable  to  introduce  into  equation  (16)  proportionality  factors  Pw  and  Pn, 
to  be  determined  from  the  observations. 

Hence,  equation  (16)  is  now  rewritten  thus  for  Lake  Erie: 

Ei  =  +  (6-8)  RWPW+  (5-7)  RnPn  =  +  (6-8)  Cw+  (5-7)  Cn  (18) 

in  which  Pw  and  Pn  are  proportionality  factors  not  necessarily  assumed  to 
be  equal,  and  C.-BJ>.  and  C.-RJ-.  (19) 

It  is  desirable  to  note  that  the  proportionality  factors  Pw  and  Pn,  to  be 
derived  from  the  observations,  tend  to  take  into  account  several  effects:  (1) 
certain  effects  arising  from  friction  and  free  oscillations  just  referred  to;  (2) 
errors  of  certain  kinds  in  assumption  No.  2,  to  which  attention  has  already 
been  called  (on  page  16).  There  may  also  be  some  tendency  for  the  wave 
produced  by  barometric  influences  to  be  modified  by  the  configuration  of 
the  shores  and  bottom  as  it  progresses  in  such  wise  that  the  wave  may  be 
accentuated  or  modified  and  given  a  larger,  or  possibly  smaller,  range  at  the 
gage  station  than  it  otherwise  would  have.  This  will  also  be  taken  into 


18  EFFECTS   OF   WINDS   AND    OF 

account  (in  part  at  least)  by  the  proportionality  factors  Pw  and  Pn  derived 
separately  from  the  observations  at  each  gage  station.  The  accentuation  or 
modification  here  referred  to  is  that  peculiar  to  the  particular  locality  at  and 
near  the  gage  station,  and  which  is  likely  to  exist  in  addition  to  the  general 
accentuation  or  modification  of  the  wave  considered  as  one  unit  for  a  whole 
lake. 

FORM  OF  OBSERVATION  EQUATIONS  FOR  BAROMETRIC 
EFFECTS. 

It  is  desirable  to  express  the  relation  between  the  mean  barometric  effects 
at  any  point  on  Lake  Erie  on  two  successive  days,  on  the  one  hand,  and  the 
barometric  difference  (6-8)  and  (5-7),  see  pages  15-16,  on  the  other  hand. 
The  development  of  the  corresponding  expression  for  Lake  Michigan-Huron 
will  be  given  later. 

It  is  proposed  to  write  one  observation  equation  for  each  day.  The  day 
to  which  the  equation  is  credited  will  be  called  the  current  day  and  the  next 
earlier  day  will  be  called  the  preceding  day.  Each  equation  is  to  express 
the  change  in  the  barometric  effect  from  the  preceding  to  the  current  day. 

Let  it  be  assumed  that  between  8  a.m.  and  8  p.m.  of  the  preceding  day 
(6-8)  increased  by  an  amount  —  bm  =  (6-8)  at  8  p.m.  minus  (6-8)  at  8  a.m., 
at  a  uniform  rate,  and  that  no  other  changes  in  barometric  gradients  occurred 
in  the  two  days. 

From  equation  (18),  on  the  assumptions  stated  and  assuming  that  there  is 
no  lag,  the  increase  in  the  elevation  of  the  water  surface  at  the  gage  station 
will  be  at  a  uniform  rate  from  8  a.m.  to  8  p.m.  and  the  total  rise  will  be 
b^Cu,.  The  variation  in  elevation  of  water  surface  at  the  gage  will  be  ex- 
pressed by  the  line  marked  "Bl  No  Lag"  on  plate  3.  Counting  from  the 
dotted  zero  line  indicated  on  the  drawing,  the  elevation  of  the  water  will  be 
zero  during  the  8  hours  from  midnight  to  8  a.m.  on  the  preceding  day,  and 
will  vary  from  zero  to  bwlCw  during  the  12  hours  from  8  a.m.  to  8  p.m.  with  a 
mean  elevation  of  0.5  bmCw  during  that  12  hours.  The  elevation  of  the 
water  surface  will  remain  bmCw  during  the  last  4  hours  of  the  preceding  day 
and  throughout  all  of  the  24  hours  of  the  current  day. 

With  reference  to  the  dotted  line  the  mean  elevation  of  the  water  surface 
on  the  preceding  day  will  therefore  be 

(0.56u,Cu,)(12)+(bu>1Ctc)4     10 

~24~~  =2l6-^ 

and  on  the  current  day  will  be  b^C^,, 

Hence,  the  increase  in  mean  elevation  for  the  current  day  over  the  mean 
elevation  for  the  preceding  day  will  be 

W7.  -  ^C.  =  1!  ft^c.  =  bVlBm  (20) 

IU-ic.  (21) 


BAROMETRIC   PRESSURES   ON   THE   GREAT   LAKES  19 

Similarly,  the  line  marked  "B2  No  Lag"  in  plate  3  represents  the  variation 
in  the  elevation  of  the  water  surface  which  will  occur  if  (6-8)  increases  by 
an  amount  —  &«*  at  a  uniform  rate  from  8  p.m.  of  the  preceding  day  to  8  a.m. 
of  the  current  day.  By  the  same  process  of  reasoning  which  was  used  above 
in  connection  with  6W,  it  may  be  shown  that  the  increase  in  mean  elevation 
for  the  current  day  over  the  mean  elevation  for  the  preceding  day  will  be 


(22} 
in  which 

B^C  (23) 

So,  too,  the  line  marked  "  B3  No  Lag"  on  plate  3  represents  the  variation 
in  elevation  if  (6-8)  increases  by  an  amount  —  bm  at  a  uniform  rate  from  8 
a.m.  of  the  current  day  to  8  p.m.  of  the  current  day.  In  this  case  the 
increase  in  the  mean  elevation  for  the  current  day  over  the  mean  elevation 
for  the  preceding  day  will  be 

(24) 


in  which 

*„-£<?„  (25) 

The  line  marked  "  B0  No  Lag"  on  plate  3  represents  the  variation  in  eleva- 
tion if  (6-8)  increases  an  amount  —  bm  at  a  uniform  rate  from  8  p.m.  of  the 
day  before  the  preceding  day  to  8  a.m.  of  the  preceding  day.  In  this  case 
the  increase  in  the  mean  elevation  for  the  current  day  over  the  mean  eleva- 
tion for  the  preceding  day  will  be 

I^C,.  =  &«.««  (26) 

in  which 

B»=^CW  (27) 

Note  that  Bw,  Bm,  Bm,  and  Bm  (see  equations  27,  21,  23,  and  25)  are 
all  constants  to  be  derived  from  the  least-square  computation.  Each  one 
involves  the  proportionality  factor  Pw  and  other  values  which  are  different 
•at  different  stations  but  do  not  vary  with  the  lapse  of  time.  Consult  equa- 
tions (19)  and  (17). 

Normally,  (6-8)  is  found  to  be  different  at  each  of  the  successive  epochs 
8  a.m.  and  8  p.m.  at  which  the  barometric  pressures  are  determined  from 
the  forecast  maps.  Hence,  on  the  assumptions  which  have  been  stated,  the 
total  decrease,  due  to  this  cause,  in  the  mean  elevation  of  the  current  day 
over  the  mean  elevation  for  the  preceding  day  will  be 

(28) 


20  EFFECTS   OF   WINDS   AND    OF 

Let  the  corresponding  notation  with  reference  to  barometric  gradients 
along  the  meridian  be  used.  That  is,  let  &„„,  bni,  bni,  and  &„,  be  the  amounts 
by  which  (5-7)  decreases  in  each  of  the  several  12-hour  periods  which  were 
specified  in  connection  with  bw,  bwi,  bwi,  and  bm,  respectively;  and  let  Bnt, 
Bni,  Bnt)  and  Bni  have  meanings  corresponding  to  those  already  specified  for 
BW>  BWI>  ^wi)  and  Bw,  respectively.  By  the  same  reasoning  which  was  used 
in  connection  with  barometric  gradients  along  parallels,  it  may  be  shown  that 
the  decrease,  due  to  change  in  barometric  gradients  along  meridians,  in  the 
mean  elevation  of  the  current  day  over  the  mean  elevation  of  the  preceding 
day  will  be  6J5m+&A+MMA,£n,  (29) 


On  the  assumptions  thus  far  stated,  the  form  of  each  observation  equation, 
one  for  each  day,  is  as  follows: 


bniBnt+I  =  V    (30) 

In  equation  (30),  7  is  the  observed  rise  in  the  elevation  of  the  water  sur- 
face at  the  gage  station  —  that  is,  the  mean  observed  elevation  for  the  current 
day  minus  the  mean  observed  elevation  for  the  preceding  day.  The  second 
member  of  the  equation  is  a  residual,  v,  which  is  the  discrepancy  between 
theory  and  observation  for  the  particular  day.  The  least-square  solu- 
tion serves  to  determine  the  most  probable  values  for  the  unknowns  Bw, 
B^  .  .  .  Bnv  Bny  These  values  are  the  ones  which  will  make  the  sum 
of  the  squares  of  the  system  of  residuals,  v,  a  minimum. 

Note  that  Bw  Bni  .  .  .  Bni,  Bn3  have  values  in  terms  of  Cw  and  Cn  which 
are  tabulated  in  the  third  line  of  table  No.  1  (page  21)  as  derived  from  equa- 
tions (27),  (21),  (23),  and  (25).  When  these  values  of  B^,  B^  .  .  .  Bw 
Bnt  have  been  determined  by  the  least-square  solution,  it  will  then  be  pos- 
sible to  compute  Cw  and  Cn. 

LAG   IN  BAROMETRIC  EFFECTS. 

Thus  far,  in  fixing  the  form  of  the  observation  equations  (30),  it  has  been 
assumed  that  there  is  no  lag  in  the  response  of  the  water  to  barometric 
changes. 

Let  it  be  assumed  that  the  response  occurs  with  a  lag  which  is  to  be  deter- 
mined from  the  observations.  The  observation  equations  will  remain  as 
before,  as  shown  in  (30),  but  certain  modifications,  which  are  about  to  be 
indicated,  will  be  necessary  in  interpreting  the  equations  and  in  interpreting 
the  derived  values  of  Bw  B^,  .  .  .  Bw  Bnt. 

Let  it  be  assumed  for  a  moment  that  the  response  of  the  water  surface  to 
barometric  changes  lags  4  hours  behind  such  changes.  Then  the  responses 
of  the  water  surface  are  properly  represented  on  the  same  basis  as  before  by 
the  lines  marked  "Blt  4hLag,"  "B2,  4hLag,"  "Bt,  4hLag,"  and  "B0,  4hLag" 
on  plate  3. 

Consider  in  detail  the  first  of  these  cases.  It  is  assumed  in  this  case  that 
between  8  a.m.  and  8  p.m.  of  the  preceding  day  (6-8)  increased  by  an  amount 


BAROMETRIC  PRESSURES  ON  THE  GREAT  LAKES 


21 


—  bwi  =  (6-8)  at  8  p.m.  minus  (6-8)  at  8  a.m.,  at  a  uniform  rate,  that  no 
other  changes  in  barometric  gradients  occurred  in  the  two  days,  and  that 
the  lag  of  the  change  in  the  water  surface  behind  the  barometric  change  is 
4  hours.  The  increase  in  the  elevation  of  the  water  surface  at  the  gage 
station  will  be  at  a  uniform  rate  from  noon  to  midnight.  The  total  rise 
will  be  bmCw.  The  rise  is  properly  shown  in  the  line  on  plate  3  marked  "  BI, 
4hLag."  Counting  from  the  dotted  zero  line  indicated  on  the  drawing,  the 
elevation  of  the  water  will  be  zero  during  the  12  hours  from  midnight  to 
noon  of  the  preceding  day  and  will  vary  from  zero  to  bmCw  during  the  12 
hours  from  noon  to  midnight  on  the  preceding  day,  with  a  mean  elevation  of 
0.5  bwlCw  during  that  12  hours.  The  elevation  of  the  water  surface  will 
remain  bwCw  throughout  all  of  the  24  hours  of  the  current  day.  With 
reference  to  the  dotted  line,  the  mean  elevation  of  the  water  surface  on  the 
preceding  day  will  therefore  be 

(0.5bmCw)12_  6& 
24  24  w 

and  on  the  current  day  will  be  bWlCw.  Hence,  the  increase  in  mean  elevation 
of  the  current  day  over  the  preceding  day  will  be 


—  —bC  = 


in  which 


18 


(31) 


(32) 


TABLE  No.  1. 


Bm  or  Bn» 

Bm  or  Bm 

Bwi  or  Bnt 

Bin  or  Bn» 

Anticipation,  or  negative  lag,  2  hours  .  . 
Anticipation,  or  negative  lag,  1  hour  .  .  . 
No  lag  

1.5 
2.0 

2.7 

12.0 
13.0 
14.0 

21.0 
20.9 
20.7 

12.0 
11.0 
10.0 

Lag,  1  hour  
Lag,  2  hours  

3.4 
4.2 

15.0 
16.0 

20.2 
19.6 

9.0 
8.0 

Lag,  3  hours  
Lag,  4  hours 

5.1 
6  0 

17.0 
18  0 

18.9 
18  0 

7.0 
6  0 

Lag,  5  hours  

7.0 

18.9 

17.0 

51 

Lag,  6  hours  .  . 

8  0 

19  6 

16  0 

4.2 

Lag,  7  hours  

9.0 

20.2 

15.0 

3.4 

Lag,  8  hours  
Lag,  9  hours 

10.0 
11  0 

20.6 
20  9 

14.0 
13  0 

2.7 
2.1 

Lag,  10  hours  

12  0 

21.0 

12.0 

1.5 

Lag,  1  1  hours  

13  0 

21  0 

11.0 

1.0 

Lag,  12  hours  

14.0 

20.7 

10.0 

.7 

Lag,  13  hours  
Lag,  14  hours   . 

15.0 
16  0 

20.3 
19  7 

9.0 
8.0 

.4 

.2 

Lag,  15  hours  

17.0 

19.0 

7.0 

.0 

Lag,  16  hours  
Lag,  17  hours 

18.0 
18  9 

18.0 
17  0 

6.0 
5.1 

.0 

22 


EFFECTS   OF   WINDS   AND    OF 


Similarly,  by  following  through  the  reasoning  corresponding  to  lines  " 
4hLag,"  "B3,  4hLag,"  and  "  B0,  4hLag"  on  plate  3  it  may  be  shown  that 


£,„.  =  — « 


(33) 


B, 


C 
24    " 


(34) 


B^—Cv  (35) 

when  the  lag  is  4  hours. 

The  various  values  of  Bm,  Bm,  .  .  .  corresponding  to  a  lag  of  4  hours 
are  shown  in  the  seventh  line  of  table  No.  1. 

The  tabular  values  are  Bwt  or  B^,  etc.,  in  terms  of  Cw  or  Cn  expressed  in 
units  of  ^  Cw  or  •&  Cn. 

The  various  lines  in  table  No.  1,  for  various  assumed  values  of  the  lag, 
have  each  been  computed  by  the  method  illustrated  by  the  examples  which 
have  been  given.  < 


Table  No.  1  makes  it  clear  that  the  values  of  Bmt  B, 


Bn,,  Br 


depend  upon  the  actual  lags  and  upon  the  actual  values  of  Cw  and  Cn.  If 
observation  equations  of  the  form  shown  in  (30)  are  used  and  the  values  of 
BU»,  BW»  -  '  -  B^,  Bn  are  derived  from  the  least-square  solution,  the  lag 
may  be  determined  from  these  values  by  the  use  of  table  No.  2,  which  has 


TABLE  No.  2. 


Bwt         fin, 
B^°T^ 

*!£!  or  *5 

Bwe          Bna 

Anticipation,  or  negative  lag,  2  hours  
Anticipation,  or  negative  lag,  1  hour  

1.75 
1.61 

8.00 
5.50 

No  lag     . 

1  48 

3  70 

Lag,  1  hour  

1  35 

2  65 

Lag,  2  hours  .  . 

1  22 

1  90 

Lag,  3  hours  

1  11 

1  37 

Lag,  4  hours  .... 

1  00 

1  00 

Lag,  5  hours  

90 

73 

Lag,  6  hours  

82 

52 

Lag,  7  hours  .  . 

74 

38 

Lag,  8  hours  

68 

27 

Lag,  9  hours  

62 

19 

Lag,  10  hours  

57 

12 

Lag,  11  hours  

52 

08 

Lag,  12  hours  . 

48 

05 

Lag,  13  hours  

44 

03 

Lag,  14  hours  

41 

01 

Lag,  15  hours 

37 

oo 

Lag,  16  hours  

33 

00 

Lag,  17  hours  

30 

BAROMETRIC   PRESSURES   ON   THE   GREAT   LAKES  23 

been  computed  directly  from  table  No.  1  by  simple  division.     For  example, 

for  no  lag  the  values  of  Bm  and  Bm  from  table  No.  1  are  — —  and  — — . 

24  24 

Hence,  in  table  No.  2,  ^  for  no  lag  is  shown  as  1.48  (=  —  Y 

Two  values  of  the  lag  may  be  determined  from  table  No.  2,  one  from  the 

r>  n  r>  T> 

ratio  — -  or  — ^  and  the  other  from  the  ratio  — —  or  — — .     The  discrepancy 
Bm        Bni  Bm        Bn, 

between  the  two  values  serves  as  a  test  of  the  degree  of  accuracy  with  which 
the  lag  is  determined  by  taking  the  mean  of  the  two  values. 

VALUES  OF  Cw  AND  Cn. 

In  table  No.  1  it  may  be  noted  that  each  of  the  following  equations  is 
either  exactly  or  nearly  true: 

5«+JB«-Cw  (36) 

*«.+*«  =  £«  (37) 

£„+£„,  =  Cn  (38) 

Bm+Bm  =  Cn  (39) 

If  table  No.  2  shows  equations  (36)  to  (39)  to  be  true  for  the  particular 
case  under  consideration,  then  Cw  and  Cn  may  be  computed  most  conven- 
iently directly  from  these  equations,  regardless  of  the  lag. 

If  the  particular  case  is  for  no  lag,  for  example,  the  table  shows  that  two 
values  of  Cw  may  be  computed,  after  said  lag  is  known,  from  the  two  equa- 
tions, written  from  table  No.  1 : 

00  A 

^+^=^CW  =  0.975CW  and  BW+B^CW 

For  the  cases  actually  encountered  in  the  investigation,  equations  (36)  to 
(39)  were  used  as  being  sufficiently  accurate. 

From  these  equations  (36)  to  (39)  two  values  of  Cw  and  two  values  of  Cn 
may  be  determined  from  each  least-square  solution.  The  discrepancy  be- 
tween the  two  values  serves  as  a  test  of  the  degree  of  accuracy  with  which 
Cw  or  Cn  is  determined  by  taking  the  mean  of  the  two  values. 

ASSUMED  UNIFORM  CHANGE  IN  BAROMETRIC  GRADIENTS. 

The  derivation  of  the  form  of  observation  equations  shown  as  equation 
(30)  on  page  20  is  based  on  an  assumption  which  is  stated  explicitly  below, 
for  convenient  reference,  as  assumption  No.  3. 

ASSUMPTION  No.  3. 

It  is  assumed  that  the  barometric  gradients  along  parallels  and  along 
meridians  vary  at  a  uniform  rate  in  each  12-hour  interval  between  the 
epochs  8  a.m.  and  8  p.m.  for  which  the  forecast  maps  show  the  facts. 


24  EFFECTS   OF   WINDS   AND   OF 

Assumption  No.  3  is  merely  an  approximation  adopted  to  simplify  the 
computation.  The  rate  at  which  any  barometric  gradient  changes  varies 
continuously  in  general.  The  errors  in  the  final  computed  results  introduced 
by  assumption  No.  3  are  believed  to  be  small.  Such  errors  are  discussed 
later  in  an  appropriate  context. 

The  preceding  statement  in  regard  to  the  derivation  of  the  form  of  the 
observation  equations  for  the  least-square  solutions  which  determine  the 
barometric  constants  is  written  primarily  with  reference  to  Lake  Erie.  The 
only  modification  necessary  to  make  the  statement  appropriate  for  Lake 
Michigan-Huron  is  that  indicated  on  pages  15-16,  where  it  is  shown  that  for 
Lake  Michigan-Huron  the  barometric  gradients  are  to  be  taken  from  ex- 
pressions (14)  and  (15)  rather  than  from  expressions  (12)  and  (13),  which 
were  used  on  Lake  Erie.  The  corresponding  changes  must  be  made  in 
equations  (16)  and  (17).  For  the  locations  of  the  points  3,  4,  5,  and  6  used 
in  expressions  (14)  and  (15),  see  page  15  and  plate  2. 


EXAMPLE  OF  OBSERVATION  EQUATIONS  FOR  BAROMETRIC 

EFFECT. 

The  form  of  the  observation  equations  for  a  least-square  solution  to  deter- 
mine barometric  effects  is  that  shown  in  equation  (30),  page  20,  which  is 
here  repeated  for  convenience  as  equation  (40) : 

*JB»+l^m+b*Bm+b^+1^m+I-  V    (40) 


The  meanings  of  the  separate  terms  in  the  equation  are  also  repeated  here 
for  convenience. 

The  current  day  is  defined  as  the  day  to  which  the  equation  is  assigned  in 
listing  the  equations. 

For  Lake  Erie : 

&u»=(6-8)  at  8  p.m.  on  the  day  before  the  day  preceding  the  current 

day  minus  (6-8)  at  8  a.m.  on  that  preceding  day. 
bun=  (6-8)  at  8  a.m.  of  the  preceding  day  minus  (6-8)  at  8  p.m.  of  that 

day. 
&u*=(6-8)  at  8  p.m.  of  the  preceding  day  minus  (6-8)  at  8  a.m.  of  the 

current  day. 
^M»==(6-8)  at  8  a.m.  of  the  current  day  minus  (6-8)  at  8  p.m.  of  the 

current  day. 

The  quantity  (6-8)  is  the  barometric  pressure  at  the  point  marked  6  on 
plate  2  minus  the  barometric  pressure  at  the  point  marked  8  on  that  plate. 
The  barometric  pressure  is  expressed  in  inches  of  mercury. 

For  Lake  Michigan-Huron,  use  (4-5)  instead  of  (6-8) ;  see  plate  2. 

The  appropriate  modification  for  any  other  lake  is  evident.  A  point  to 
the  westward  of  the  lake  is  to  be  used  in  the  place  of  the  point  6,  and  one  to 
the  eastward  and  in  the  same  parallel  is  to  be  used  in  the  place  of  point  8. 


BAROMETRIC   PRESSURES   ON   THE    GREAT   LAKES  25 

For  Lake  Erie: 

&no=(5-7)  at  8  p.m.  on  the  day  before  the  day  preceding  the  current 

day  minus  (5-7)  at  8  a.m.  on  that  preceding  day. 
6ni=  (5-7)  at  8  a.m.  of  the  preceding  day  minus  (5-7)  at  8  p.m.  of  that 

day. 
ft/* =  (5-7)  at  8  p.m.  of  the  preceding  day  minus  (5-7)  at  8  a.m.  of  the 

current  day. 
&n,=  (5-7)  at  8  a.m.  of  the  current  day  minus  (5-7)  at  8  p.m.  of  the 

current  day. 

The  quantity  (5-7)  is  the  barometric  pressure  at  the  point  marked  5  on 
plate  2  minus  the  barometric  pressure  at  the  point  marked  7  on  that  plate. 

For  Lake  Michigan-Huron,  use  (3-6)  instead  of  (5-7) ;  see  plate  2. 

The  appropriate  modification  for  any  other  lake  is  to  use  in  place  of  point 
5  (for  Lake  Erie)  a  point  to  the  northward  of  the  lake,  and  a  point  to  the 
southward  of  the  lake  and  in  the  same  meridian  in  the  place  of  point  7. 

BW>,  Bw,  Bw,  BW,  Bw  Bni,  Bni,  and  B^  are  constants  of  which  the  values 
are  to  be  determined  from  the  least-square  solution. 

/  is  the  observed  rise  in  the  water  surface,  at  the  gage  station  from  the 
preceding  day  to  the  current  day,  corrected  for  wind  effects  if  such  correc- 
tions are  available,  and  corrected  for  inflow  from  the  next  lake  above,  out- 
flow to  the  next  lake  below,  and  rainfall  on  the  lake  surface.  In  other  words, 
ignoring  the  various  corrections  for  the  moment,  /  is  the  mean  elevation  of 
the  water  surface  on  the  current  day  minus  the  mean  elevation  of  the  water 
surface  on  the  preceding  day. 

The  residual  v  in  the  second  member  of  equation  (40)  is  to  be  derived 
by  substitution  of  the  computed  values  of  Bm,  Bm,  .  .  .  Bw  #m  in  the 
observation  equations  after  the  solution  is  complete.  The  residual  v,  one  for 
any  observation  equation,  is  the  discrepancy  between  the  theory  and  the 
observed  facts  for  the  particular  day. 

The  following  set  of  observation  equations  for  Milwaukee  for  the  month 
of  September  1910  serve  as  a  typical  illustration.  They  are  a  part  of  the 
equations  for  solution  Kz,  which  included  in  all  186  such  equations  covering 
the  months  June  to  September  1910  and  June  to  September  1911 — 8  months 
in  all. 

No  record  of  the  elevation  of  the  water  surface  was  obtained  from  the 
gage  at  Milwaukee  on  September  2,  3,  and  4. 

The  following  possible  equations  were  rejected:  for  September  2-5  (as  one 
equation),  for  September  7,  for  September  8,  for  September  17,  for  Septem- 
ber 24,  for  September  25,  for  September  26,  and  for  September  28.  It  is 
unusual  for  so  many  rejections  to  occur  in  any  one  month.  As  shown,  the 
equations  for  certain  days  were  combined  in  pairs  to  make  one  equation  in 
each  case,  namely,  for  August  31  and  September  1  and  for  September  12  and 
13.  Each  combined  equation  was  obtained  by  adding  corresponding  terms 
of  the  separate  equations. 


26  EFFECTS   OF   WINDS   AND    OF 

OBSERVATION  EQUATIONS. 
Date 
1910 
Sept. 
(Aug.)    31-1 +  5Bm—28Bwi  —  12Bwi+  QBw*+23Bn<>  +  6Bm—23Bnt+  3Bm-     9  =  vi 

9 +  3Bw,— 36-Btw—  QBw*+2QBwi+  5Bno  —  lQBm  +  !Bnt+  9fi»3  +  21=v» 

14 +  3Bw0+  4Bwi—  lBv>i+  5Bwi+  7Bm>+  QBm+WBnz—  £Bm  —  ll2=vi 

16 +  3Bw+  7Bwi-  2BwI+10Bw3+  3Bno+  QBm+  3Bn2-  3Bm-     3  =  vt 

19 —  21fitro  +  19Btoi  — 

20 —  3Bw»—  3Bwi  — 

21 —  9-Bwo—  4Btci  — 

22....-  IBw,' 

27. ...  +l3Bw,+13Bm-20Bwt-15Bwt- 

29 +  7Bw,  +  10Bwi+  7Bwt+  2Bwt+  3Bn*-12Bni  + 

30....+  7Bw,+ 


8JBn,+  68=t>u. 


The  basis  on  which  rejections  and  combinations  were  made  is  shown  later 
in  connection  with  the  discussion  of  the  accuracy  of  the  computed  baro- 
metric effects. 

BAROMETRIC  TERMS  IN  OBSERVATION  EQUATIONS. 

For  Lake  Michigan  the  barometric  pressures  used  in  the  computations 
were  those  read  at  points  3,  4,  5,  and  6  as  shown  on  plate  2.  The  values  of 
the  barometric  pressures  at  these  points  at  8  a.m.  and  8  p.m.  (75th  meridian 
time)  of  each  day  for  a  part  of  September  1910  are  shown  in  table  No.  3. 
The  table  also  shows  the  values  of  the  differences  of  barometric  pressure 
(4-5)  and  (3-6)  which  were  used  in  the  preparation  of  a  part  of  the  above 
observation  equations. 

In  the  above  equations  the  values  of  bwi,  bw  .  .  .  6nj  and  6W  are  ex- 
pressed in  units  of  0.01  inch,  and  the  absolute  term  7,  the  rise  of  the  water 
surface,  is  expressed  in  units  of  0.001  foot.  This  arbitrary  selection  of 
units  was  made  in  order  that  the  average  magnitude  of  quantities  in  the 
various  columns,  as  the  equations  are  arranged  in  the  example  given,  might 
be  about  the  same.  To  have  them  about  the  same  increases  the  accuracy 
of  certain  checks  against  error  in  the  computations. 

The  derivation  of  the  values  of  bm,  bmt  .  .  .  bn,  and  &„,  as  given  in  the 
observation  equations  may  be  verified  from  the  table  for  the  period  which  it 
covers.  For  example,  note  that  the  exceptionally  large  value  of  bm  in  the 
equation  for  September  9  is  (4-5)  at  8  a.m.  of  September  8,  (  —  17),  minus 
(4-5)  at  8  p.m.  of  September  8,  (+19),  or  -36.  This  is  the  most  rapid 
change  of  barometric  difference  shown  anywhere  in  table  No.  3.  Note  that 


BAROMETRIC  PRESSURES  ON  THE  GREAT  LAKES 


27 


there  was  also  a  rather  rapid  change  in  the  barometric  difference  in  the 
meridian,  difference  3-6,  between  8  a.m.  and  8  p.m.  on  September  8  from 
which  the  term  bn\  in  the  September  9  observation  equation  is  derived. 


TABLE  No.  3. 


Date  1910. 

Hour. 

Barometric  pressure  in  inches  of 
mercury. 

In  unite  of  0.01 
inch. 

At 
point  3. 

At 
point  4. 

At 
point  5. 

At 
point  6. 

Diff. 
(4-5) 

Diff. 
(3-6) 

Sept     1 

8  a.m. 
8  p.m. 
8  a.m. 
8  p.m. 
8  a.m. 
8  p.m. 

8a.m. 
8  p.m. 
8  a.m. 
8  p.m. 
8  a.m. 
8  p.m. 

8  a.m. 
8p.m. 
8  a.m. 
8  p.m. 
8  a.m. 
8  p.m. 

8  a.m. 
8p.m. 
8  a.m. 
8  p.m. 
8  a.m. 
8  p.m. 

8  a.m. 
8  p.m. 
8  a.m. 
8  p.m. 
8a.m. 
8  p.m. 

30.32 
30.23 
30.19 
29.83 
29.70 
29.85 

29.90 
29.93 
29.92 
29.89 
29.70 
29.93 

30.19 
30.02 
29.90 
29.99 
30.20 
30.13 

30.12 
30.03 
30.02 
30.03 
30.21 
30.37 

30.45 
30.31 
30.30 
30.28 
30.30 
30.21 

30.30 
30.18 
30.13 
29.80 
29.76 
29.82 

29.85 
29.83 
29.77 
29.78 
29.78 
30.03 

30.15 
30.04 
29.94 
30.08 
30.28 
30.18 

30.18 
29.99 
30.02 
29.25 
30.22 
30.34 

30.44 
30.36 
30.40 
30.31 
30.35 
30.26 

30.19 
30.16 
30.23 
29.96 
29.73 
29.74 

29.95 
29.95 
29.91 
29.89 
29.80 
29.90 

30.13 
30.18 
30.11 
29.89 
30.03 
30.13 

30.24 
30.14 
30.20 
30.10 
30.12 
30.21 

30.34 
30.30 
30.33 
30.29 
30.36 
30.34 

30.20 
30.14 
30.16 
29.87 
29.77 
29.80 

29.85 
29.84 
29.77 
29.80 
29.88 
30.04 

30.23 
30.16 
30.09 
29.99 
30.21 
30.23 

30.26 
30.09 
30.12 
29.99 
30.04 
30.22 

30.37 
30.32 
30.41 
30.35 
30.40 
30.31 

+  11 
+  2 
-10 
-16 

+  3 
+  8 

-10 
-12 
-14 
-11 
-  2 
+  13 

+  2 
-14 
-17 
+19 
+25 
+  5 

-  6 
-15 
-18 
-15 
+  10 
+  13 

+  10 
+  6 
+  7 
+  2 
-  1 
g 

+12 
+  9 
+  3 
-  4 
-  7 
+  5 

+  5 
+  9 
+15 
+  9 
-18 
-11 

-  4 
-14 
-19 
0 
-  1 
-10 

-14 
-  6 
-10 

+  4 
+17 
+15 

+  8 
-  1 
-11 
-  7 
-10 
-10 

1  
2 

2  

3  
3  

4  
4  
5  
5  

6           

6 

7 

7  

8             .    ... 

8  

9   

9  

10 

10  

11           

11  

12  
12 

13 

13  

14  
14 

15  
15  

Table  No.  4  shows  the  observed  elevations  referred  to  mean  sea-level  of 
the  water  surface  at  the  Milwaukee  gage,  the  observed  rise  of  water  surface 
for  each  day,  the  corrections  to  observed  rise  for  known  inflow,  outflow,  and 
rainfall  on  the  lake  surface,  the  corrections  to  the  observed  rise  for  wind 
effects,  and  the  corrected  rise  as  used  in  the  observation  equations  as  the  I 
term. 

3 


28  EFFECTS   OF   WINDS   AND    OF 

CHANGE  OF  ELEVATION  USED  IN  OBSERVATION  EQUATIONS. 

Each  observed  daily  elevation  is  the  mean  of  24  hourly  elevations  obtained 
from  the  gage  record. 

The  inflow  to  Lake  Michigan-Huron,  through  the  St.  Marys  River,  from 
Lake  Superior  was  determined  each  day  by  the  U.  S.  Lake  Survey  by  means 
of  a  recording  gage  giving  the  elevation  of  the  water  surface  at  Sault  Ste. 
Marie  above  the  rapids.  Similarly,  the  outflow  from  Lake  Michigan- 
Huron  to  Lake  Erie  was  determined  for  each  day  from  the  recorded  elevation 
of  the  water  surface  at  Fort  Gratiot  and  at  St.  Clair  Flats  in  the  St.  Clair 
River.  In  each  of  these  cases  the  recorded  elevations  of  water  surface  were 
converted  to  volume  of  stream-flow  by  means  of  the  known  relations  between 
these  quantities  which  had  been  established  by  river  gagings  made  by  the 
U.  S.  Lake  Survey.  As  these  corrections  for  inflow  and  outflow  are  small 
and  are  abundantly  accurate  for  the  purpose,  the  explanation  in  detail  of 
their  computation  is  not  here  given. 

No  correction  was  applied  for  run-off  into  Lake  Michigan-Huron  from  the 
surrounding  land-drainage  area.  This  quantity  was  very  difficult  to  deter- 
mine at  the  time  this  investigation  was  made.  So,  too,  no  correction  is 
applied  for  evaporation,  for  a  similar  reason. 

The  amount  of  rise  of  water  surface  of  the  lake  on  each  day,  produced  by 
rainfall  on  the  lake  surface,  was  computed  from  a  number  of  rain  gages 
operated  by  the  U.  S.  Weather  Bureau  and  the  Weather  Service  of  Canada. 

For  convenience  the  three  corrections — for  inflow,  for  outflow,  and  for 
rainfall  on  the  lake  surface — were  grouped  together  in  the  computation  as 
shown  in  the  third  column  of  table  No.  4.  These  three  corrections  are 
given  separately  in  the  last  three  columns  of  the  table. 

Note  that  the  difference  between  outflow  and  inflow,  or  what  might  be 
called  net  outflow,  is  no  more  than  0.008  foot  on  any  day  in  September  1910. 
Note  that  the  maximum  rise  or  fall  during  any  day  of  the  month  due  to 
combined  inflow,  outflow,  and  rainfall  on  the  lake  surface  was  only  0.033 
foot.  It  is  clear,  after  comparing  these  values  with  the  observed  rise  of  the 
lake  surface  at  Milwaukee  for  each  day,  as  shown  in  table  No.  4,  that  the 
observed  rise  is  evidently  due  to  causes  other  than  these  three,  which  are 
minor  in  their  effect  upon  the  daily  fluctuations  of  level  as  compared  with 
other  causes  which  are  operating. 

The  corrections  for  wind  effects  as  shown,  at  a  maximum  only  0.001  foot 
for  September  24  and  September  25,  were  computed  by  methods  shown  in 
detail  later  in  this  publication. 

In  all  columns  except  the  second  the  change  shown  is  for  that  from  the 
preceding  day  to  the  day  indicated  in  the  first  column  on  that  line.  For 
example,  —0.180  foot  is  the  rise  of  water  surface  from  September  5  to  Septem- 
ber 6,  namely,  580.07-580.25.  Similarly,  —  0.033  foot  is  the  correction  for 
rainfall,  inflow,  and  outflow  combined  from  September  5  to  September  6. 

The  observation  equations  were  arranged  in  groups  of  one  month  each. 


BAROMETRIC   PRESSURES  ON   THE    GREAT  LAKES 


29 


The  groups  were  studied  separately  as  well  as  in  combination  with  each 
other  as  a  single  set  of  equations  for  one  complete  solution. 

TABLE  No.  4— For  Milwaukee,  Unit  0.001  foot. 


Observed 
eleva- 
tion of 
water 
surface. 

Rise  of 
water 
surface. 

Correc- 
tion for 
rainfall, 
inflow 
and 
outflow. 

Correc- 
tion for 
wind 
effect. 

Cor- 
rected 
rise. 

Correc- 
tion for 
rainfall. 

Correc- 
tion for 
inflow. 

Correc- 
tion for 
outflow. 

1910 

feet. 

Sept.   1... 

580.32 

.... 

+  7 

0 

.... 

-  1 

-5 

+13 

2... 

+  7 

0 

0 

-5 

+12 

3... 

.... 

+  1 

0 

.... 

-  7 

-5 

+13 

4... 

-  2 

0 

-10 

-5 

+13 

5... 

58()!25 

-'76 

-26 

0 

.... 

-33 

-5 

+12 

6... 

.07 

-180 

-33 

0 

-213 

-40 

-5 

+12 

7... 

.00 

-  70 

-  2 

0 

-  72 

-10 

-5 

+  13 

8... 

.23 

+230 

+  2 

0 

+232 

-  6 

-5 

+13 

9... 

.25 

+  20 

+  1 

0 

+  21 

-  7 

-5 

+13 

10... 

.25 

0 

+  7 

0 

+     7 

-  1 

-5 

+  13 

11..  . 

.29 

+  40 

+  5 

0 

+  45 

-  2 

-5 

+  12 

12... 

.51 

+220 

-19 

0 

+201 

-27 

-5 

+13 

13... 

.37 

-140 

-17 

0 

-157 

-25 

-5 

+  13 

14... 

.25 

-120 

+  8 

0 

-112 

0 

-5 

+  13 

15... 

.31 

+  60 

+  8 

0 

+  68 

0 

c 

+13 

16... 

.30 

-  10 

+  7 

0 

-     3 

0 

-5 

+  12 

17... 

.16 

-140 

-  2 

0 

-142 

-  9 

-5 

+  12 

18... 

.44 

+280 

-  3 

0 

+277 

-11 

-5 

+13 

19... 

.40 

-  40 

+  5 

0 

-  35 

-  2 

-5 

+12 

20... 

.23 

-170 

+  7 

0 

-163 

0 

-5 

+  12 

21... 

.34 

+110 

+  8 

0 

+118 

0 

-5 

+  13 

22... 

.40 

+  60 

+  8 

0 

+  68 

0 

-4 

+12 

23... 

.51 

+110 

0 

0 

+110 

-  7 

-5 

+12 

24... 

.71 

+200 

-20 

-1 

+  179 

-27 

-4 

+11 

25... 

.46 

-250 

-15 

+1 

-264 

-22 

-5 

+12 

26.. 

.37 

-  90 

+  3 

0 

-  87 

-  5 

-4 

+12 

27.. 

.28 

-  90 

-  2 

0 

-  92 

-  9 

-5 

+12 

28.. 

.14 

-140 

+  3 

0 

-137 

-  5 

-5 

+13 

29.. 

.24 

+  100 

+  8 

0 

+  108 

0 

-4 

+  12 

30.. 

.12 

-120 

+  7 

0 

-113 

0 

-5 

+  12 

EXAMPLE  OF  NORMAL  EQUATIONS  FOR  BAROMETRIC 
EFFECTS. 

In  each  solution  a  set  of  normal  equations  was  first  formed  for  each  month, 
in  the  usual  way,  from  the  observation  equations  for  that  month.  These 
sets  of  normal  equations  were  then  available  for  separate  study  for  each 
month. 


30  EFFECTS   OF   WINDS   AND   OF 

These  monthly  normal  equations  were  later  combined  by  addition, 
term  by  term,  to  form  the  final  set  of  normal  equations  for  the  solution. 

The  final  normal  equations  for  solution  K%  are  given  below  as  table  No.  5. 
Solution  KI  served  to  give  the  final  determination  of  the  barometric  effects 
at  Milwaukee. 

TABLE  No.  5 — Normal  Equations,  Solution  K2,  Milwaukee. 
+179445*0-  2552Bw1-  39555**-  24475*.,+  48025n0-     5735m  +  28175n2+     3655ns- 17427  =  0 

-  2552fi.ro +  163365«n  +  33585**-  46225*,,-  33045no+  41305m  +  20245^-  14825n, +41911  =0 

-  39555«x>+  33585«,1+220355«,2-  31475**-  49165n0-  11815m  +  42695^+  27745n,+ 53880  =  0 

-  24475«,0-  46225**-  31475«-2+151045*,3-     3475no-  14385m-  42455nj  +  29045n3+ 11035  =  0 
+  48025*,0-  33045m-  49165*-2-     3475*>3+134445n0-  14375m-  33005^-  40655na-  8746=0 

-  5735wo+  41305«i-  11815**-  14385*>3-  14375n0+ 12 1495m-     4575^-       655n3-61166  =  0 
+  28175u.0+  20245*,!+  42695**-  42455**-  33005no-     4575ni  +  172065n2-  15865n3- 73936  =  0 
+     3655««-  14825^,+  27745trz+  29045**-  40655n0-       655m-  15865n2  +  101685n3+  2900  =  0 

These  normal  equations  depend  upon  observations  extending  over  220 
days,  which  were  expressed  in  186  observation  equations.  Some  of  the 
observation  equations  covered  two  or  more  days  each. 

The  solution  of  these  normal  equations  gave  the  following  values  for 
the  unknowns,  expressed  in  the  units  used  in  the  observation  equations: 
Bw0=-l.7S        £n0=+1.99 
5wi=-4.34        £nl=+6.53 
£u.2=-2.92        Bn2=+6.34 
Bw3=-   .90        Bn3=+2.03 

EXAMPLE  OF  SUBSTITUTION  IN  OBSERVATION  EQUATIONS 
FOR  BAROMETRIC  EFFECTS. 

The  above  values  by  substitution  in  the  observation  equations  for  Septem- 
ber 1910  as  given  on  page  26  gave  the  equations  which  served  to  determine 
the  residuals  v  for  this  particular  month.  From  such  residuals  the  probable 
errors  were  computed.  These  residuals  are  the  discrepancies  between  the 
theory  on  which  the  observation  equations  were  based,  on  the  one  hand,  and 
the  observed  facts,  on  the  other  hand. 

In  the  following  tabular  arrangement  of  the  substitution  in  observation 
equations  the  heading  on  each  column  identifies  the  term  of  the  observation 
equations  as  shown  on  page  26.  As  a  specific  example,  note  that  the  value 
given  in  the  Bwi  column  for  September  9,  namely,  +156,  is  the  product  of  the 
quantity  —36  in  the  September  9  observation  equation,  which  is  the  coeffi- 
cient of  BU,I  times  the  final  value  of  Bwi,  namely,  —4.34. 

The  values  of  v,  the  residuals,  were  obtained  by  adding  all  terms  in  the 
first  member  of  the  equation.  If  the  agreement  between  theory  and  ob- 
servation were  perfect,  all  i>'s  would  be  zero. 

Note  that  there  were  8  days  in  September  1910  when  the  observed  rise 
(or  fall)  at  Milwaukee,  corrected  for  rainfall,  inflow,  outflow,  and  wind,  was 
more  than  0.1  foot  (corresponding  to  100  in  the  I  column  in  the  following 
substitution  in  observation  equations),  namely,  September  6,  14,  18,  20,  21,. 


BAROMETRIC   PRESSURES   ON   THE    GREAT   LAKES  31 

23,  29,  and  30.  The  largest  residual  v  for  any  one  of  these  8  days  is  +0. 079 
foot  on  September  18.  For  that  day  the  /  term  or  observed  rise  (with  the 
small  noted  corrections)  as  expressed  by  the  /  term  was  +0.277  foot.  In 
other  words,  of  the  unusually  large  observed  rise  of  0. 280  foot  (see  table  No. 
4)  from  September  17  to  September  18,  0.003  foot  is  accounted  for  by  rain- 
fall, inflow,  and  outflow,  0.198  foot  is  accounted  for  as  barometric  effects,  and 
only  0.079  foot,  or  only  28  per  cent  of  the  observed  rise,  is  left  unaccounted 
for. 

Substitution  in  Observation  Equations  for  September  1910,  at  Milwaukee. 
The  unit  is  0.001  foot. 


Date 
31-  1 

—  9 

Bwl 
+122 

+35 

Bwt 

—  8 

+46 

Bni 

+  20 

AM 

—  127 

+  6 

/           V 

—  9  =  +76 

6.  ... 
9 

-  4 
...       —  5 

+  13 
+156 

+26 
+18 

+13 
—  18 

-12 
+10 

+  39 
—  124 

+171 
+    6 

-14 

+18 

-213  =+19 
+  21  =  +82 

10. 

+11 

-  87 

-32 

-  8 

+  2 

+  59 

+  25 

—  16 

+  7  =—39 

11 

-20 

—  39 

—  9 

+  3 

+  8 

—  52 

+  25 

—28 

+  45=  —67 

12-13 

—  5 

+  13 

+73 

—  4 

+  8 

—  78 

—  38 

4-18 

+  44=  +31 

14.... 
15.... 
16.... 
18 

-  5 
+  2 
-  5 
—  9 

-  17 
-  22 
-  30 
+  43 

+  3 
-  9 
+  6 
+61 

-  4 
-  6 
-  9 
—  17 

+14 
+20 
+  6 

+32 

+  59 
-  26 
0 
—  131 

+  63 
+  19 
+  19 
—  197 

JQOOCOC 
-I  I  4 

-112=-  7 

+  68  =  +46 
-  3=  -22 
+277  =  +79 

19 

+37 

-  82 

+  9 

+  3 

-62 

+  65 

+  32 

+16 

—  35=  -17 

20.  .  .  . 
21 

+  5 
+16 

+  13 

+  17 

+26 
+  3 

+  4 
—  8 

+  10 

+22 

+  52 
—  59 

+  70 
—  89 

-18 

+  8 

-163=-  1 

+118  =  +28 

22...  . 
23.... 
27.  .  .  . 
29 

+2 
-28 
......     -23 
—  12 

-  39 
-  26 
-  56 
—  43 

-47 

+53 
+58 
—20 

-  5 
+  2 
+13 
—  2 

-28 
+10 

-  8 
+  6 

+  26 
+  52 
-  13 

—  78 

+  32 
-101 
+171 
+  57 

+16 
-24 
-10 

+  2 

+  68  =  +25 
+110=  +48 
-  92  =+40 
+108  =+18 

30.  ... 

-12 

-     9 

-18 

+  9 

+18 

+  7 

+  76 

-  2 

-113=  -44 

THE   FIVE   FINAL   SOLUTIONS   FOR   BAROMETRIC   EFFECTS. 

The  principal  facts  for  each  of  the  five  least-square  solutions  which  served 
to  give  the  adopted  values  for  the  quantities  B^,  B^,  .  .  .  #„,,  and  B^ 
expressing  the  barometric  effects  at  the  five  stations,  Buffalo,  Cleveland, 
Milwaukee,  Harbor  Beach,  and  Mackinaw,  are  here  brought  together  for 
convenient  reference. 

Each  of  the  five  solutions  was  based  upon  8  months  of  observation  of 
water  elevations  at  the  station  named.  At  Buffalo  and  Cleveland,  the 
months  were  August  to  October,  inclusive,  1909,  and  June  to  October, 
inclusive,  1910.  At  Milwaukee,  Harbor  Beach,  and  Mackinaw,  the  months 
were  June  to  September,  inclusive,  in  each  of  the  two  years  1910  and  1911. 

In  table  No.  6  the  probable  errors  included  in  parentheses  are  estimated 
on  the  basis  of  those  which  are  not  so  marked.  These  latter  were  computed 
rigorously  from  the  residuals  and  the  normal  equations. 

The  values  of  B^,  B^,  .  .  .  B^  and  £„,  as  given  in  table  No.  6 
correspond  to  the  units  arbitrarily  adopted  for  convenience  in  the  obser- 
vation equations — that  is,  if  &«*,  6^,  .  .  .  &„,  and  b»t  are  expressed  in 
units  of  0.01  inch  and  the  above  values  for  B^,  BWl,  .  .  .  #«,  and  £„,  are 
used,  the  resulting  products  bw,  B^,  bWl  BWl  .  .  .  &„,  B^  and  6n,  Bni  are 
barometric  effects  expressed  in  units  of  0.001  foot. 


32 


EFFECTS   OF   WINDS   AND    OF 


COMPUTATION  OF  HOURLY  BAROMETRIC  EFFECTS. 

To  secure  a,  basis  for  computing  the  hourly  barometric  effects  at  a  gage 
station,  one  must  first  compute  the  lag  in  each  component  of  the  barometric 
effect  by  the  use  of  table  No.  2,  page  22,  and  compute  the  values  of  Cv  and 
Cw  by  formulae  (36),  (37),  (38),  and  (39)  of  page  23. 

At  Milwaukee  these  computations  were  made  from  the  values  shown  in 
table  No.  6. 


BAROMETRIC   PRESSURES   ON   THE    GREAT   LAKES 
The  lag  in  Cw  was  found  thus  : 


33 


B 


-2.92  __          £W3      -0.90 

+0  .  67  and  -  =  -  =  +0  .  50 


-4.34 


BVO     -1.78 


From  the  first  of  these,  according  to  table  No.  2,  the  lag  is  8.2  hours, 
and  from  the  second  it  is  6.  1  hours.  The  mean  (7  hours)  was  adopted  as 
the  most  probable  value  of  the  lag  in  Cv. 

Similarly,  the  two  values  found  for  the  lag  in  Cn  were  4.3  and  4.1  hours. 
The  adopted  value  was  taken  as  4  hours. 

The  values  of  Cw  computed  from  formulae  (36)  and  (37)  were,  respectively, 
-1.78-2.  92=  -4.  70  and  -4.34-0.90=  -5.24.  The  mean  -4.97  was 
adopted  as  the  most  probable  value  of  Cw. 

The  two  values  of  Cn  computed  from  formulae  (38)  and  (39),  respectively, 
were  +8.33  and  +8.56.  The  mean  +8.44  was  adopted  as  the  most 
probable  value  of  Cn. 

The  computations  of  lag  and  of  Cw  and  Cn  were  made  for  all  five  stations 
from  the  values  of  B^,  B^,  .  .  .  #„,,  and  B^,  shown  in  table  No.  6,  with 
the  results  shown  below  in  table  No.  7. 


TABLE  No.  7. 


Buffalo. 

Cleveland. 

Milwaukee. 

Harbor 
Beach. 

Mackinaw. 

Lag  in  Cw,  hours  

-  1* 

4 

7 

6 

-2* 

Lag  in  Cn,  hours  
Cw,  in  same  units  as  table  No.  6 
Cn,  in  same  units  as  table  No.  6 

3 

+  4.72 
-15.62 

2 

+2.12 
+6.80 

4 
-4.97 
+8.44 

6 
+6.94 
-0.44 

-1* 
-2.54 
-4.38 

*Note  that  these  three  values  of  the  lag  are  marked  with  minus  signs, 
used.     They  are  anticipations  rather  than  lags. 


They  are  to  be  so 


From  these  values  the  barometric  effect  at  each  hour  at  each  station  may 
be  computed  from  the  following  formulae,  each  of  which  is  formula  (18),  page 
17,  modified  (a)  to  adapt  it  to  the  particular  lake,  (6)  to  take  into  account 
the  lags  which  are  now  known,  and  (c)  to  adapt  it  to  the  units  used  above  in 
table  No.  7  and  in  the  observation  equations. 

In  the  following  formulae  the  quantities  (6-8),  (5-7),  (4-5),  and  (3-6)  are 
expressed  in  units  of  0.01  inch.  The  computed  effect,  Ei,  is  obtained  from 
the  formulas  in  units  of  0.001  foot. 


For  Buffalo,  E,,  for  any  hour=(6-8)  (+4.72)  +  (5-7)  (-15.62),  in 
which  (6-8)  must  be  taken  for  1  hour  later  and  (5-7)  for  3 
hours  earlier  than  the  hour  for  which  the  effect  is  being 
computed. 


(41) 


34  EFFECTS   OF   WINDS   AND   OF 

For  Cleveland,  El}  for  any  hour  =(6-8)  (+2. 12)  +  (5-7)  (+6. 80),  in         (42) 
which  (6-8)  must  be  taken  for  4  hours  earlier  and  (5-7) 
for  2  hours  earlier  than  the  hour  for  which  the  effect  is 
being  computed. 

For  Milwaukee,  Ei,  for  any  hour=(4-5)  (-4.97)  +  (3-6)  (+8.44),         (43) 
in  which  (4-5)  must  be  taken  for  7  hours  earlier  and  (3-6) 
for  4  hours  earlier  than  the  hour  for  which  the  effect  is 
being  computed. 

For    Harbor    Beach,    Elt    for    any    hour  =  (4-5)  (+6. 94) +  (3-6)         (44) 
(—0.44),  in  which  (4-5)  must  be  taken  for  6  hours  earlier 
and  (3-6)  for  6  hours  earlier  than  the  hour  for  which  the 
effect  is  being  computed. 

For  Mackinaw,  Elt  for  any  hour  =  (4-5)  ( -  2 . 54)  +  (3-6)  (-4.38),         (45) 
in  which  (4-5)  must  be  taken  for  2  hours  later  and  (3-6) 
for  1  hour  later  than  the  hour  for  which  the  effect  is  being 
computed. 

The  example  on  page  35  shows  the  details  of  the  computation  of  hourly 
barometric  effects  at  Milwaukee  on  September  24,  1910.  The  barometric 
effects  at  Milwaukee  on  this  day  were  unusually  large  and  were  changing 
with  unusual  rapidity. 

It  was  especially  desirable  to  secure  the  hourly  barometric  effects  on 
September  24  with  as  great  accuracy  as  possible,  because  this  was  one  of  the 
dates  used  in  a  computation  of  wind  effects  at  Milwaukee.  Hence,  a 
special  study  of  the  forecast  maps  of  this  and  adjacent  days  was  made  with  a 
view  to  determining  the  time  of  maximum  and  minimum  pressure  at  each 
reading  point  and  the  value  of  each  such  maximum  or  minimum.  Within 
the  limits  shown  in  the  example  there  are  six  points  found  by  this  special 
study,  namely: 

A  maximum  of  30.34  at  9  p.m.  on  Sept.  23  at  point  3, 
A  maximum  of  30. 10  at  9  p.m.  on  Sept.  23  at  point  6, 
A  maximum  of  30.26  at  3  a.m.  on  Sept.  24  at  point  5, 
A  minimum  of  29.80  at  11  a.m.  on  Sept.  24  at  point  6, 
A  minimum  of  29.80  at  5  p.m.  on  Sept.  24  at  point  4, 
A  minimum  of  29.79  at  11  p.m.  on  Sept.  24  at  point  3. 

Each  of  these  maxima  and  minima  is  inclosed  in  a  parenthesis  in  the 
example  of  computation  given. 

Consult  plate  2  for  the  location  of  the  points  3,  4,  5,  and  6  for  which  the 
barometric  pressures  were  read  from  the  forecast  maps. 

Opposite  each  maximum  or  minimum  the  pressure  as  given  in  square 
brackets  for  the  companion  point  was  obtained  by  interpolation  between  the 
next  preceding  and  next  following  value  at  that  point,  on  the  assumption 
that  during  the  interval  the  pressure  changed  at  a  uniform  rate. 

The  values  in  the  columns  (3-6)  and  (4-5)  which  are  not  included  in 
square  brackets  were  obtained  directly  by  subtraction  from  the  preceding 


BAROMETRIC    PRESSURES   ON   THE    GREAT   LAKES  35 

columns.  The  values  in  these  columns  which  are  inclosed  in  square  brackets 
were  obtained  by  interpolation  from  the  unbracketed  values  in  the  same 
columns,  on  the  assumption  that  the  rate  of  change  of  the  barometric 
gradient  was  constant  during  each  interval  between  unbracketed  values. 

Example  of  computation  of  hourly  barometric  effects — Milwaukee. 


Date  and 
hour. 

Barometric  pressure,  at  point. 

N-S 
effect. 

E-W 

effect. 

Ei 
total 
effect. 

3. 

6. 

4. 

5. 

(3-6). 

(4-5). 

Sept.  23. 
8  p.m  
9 

inches. 

30.34 

(30.34) 

inches. 

30.09 
(30.10) 

inches. 
30.20 

inches. 
30.20 

inches. 

+  .25 
+  .24 
[+.25] 
[+.27] 
[+.28] 

[+.30] 
[+.31] 
[+.33] 
[+.34] 
[+.36] 
[  +  .37] 
[+.38] 
+  .40 
[+.38] 
[+.35] 
+  .33 
(+.29] 
[+-24] 
[+.20] 
[+.16] 
[  +  .11] 
[+-07] 
[+.03] 
[-.02] 
-.06 
[-.10] 
[--14] 
-.17 
[-.17] 

inches. 

+  .00 
[-.02] 
[-.05] 
[-.08] 
[-.10] 

[-.12] 
[-.16] 
-.18 
[-.19] 
[-.20] 
[-.21] 
[-•22] 
-.23 
[--23] 
[-.22] 
[-.22] 
[--21] 
[--21] 
[-.20] 
[-.20] 
[-.19] 
-.19 
[-.16] 
[--12] 
-.08 
[-.07] 
[--06] 
[-.05] 
[-.04] 

feet. 

feet. 

feet. 

10          

11  p.m. 

M  

+  .20 
+  .21 

+  .23 
+  .24 
+  .25 
+  .26 
+  .28 
+  .29 
+  .30 
+  .31 
+  .32 
+  .34 
+  .32 
+  .30 
+  .28 
+  .24 
+  .20 
+  .17 
+  .14 
+  .09 
+  .06 
+  .03 
-.01 
-.05 

.00 
.00 
.00 
+  .01 
+  .02 
+  .04 
+  .05 
+  .06 
+  .08 
+  .09 
+  .09 
+  .10 
+  .10 
+  .11 
+  .11 
+  .11 
+  .11 
+  .11 
+  .10 
+  .10 
+  .10 
+  .10 
+  .09 
+  .09 
Mean. 

+  .20 
+  .21 
+  .23 
+  .25 
+  .27 
+  .30 
+  .33 
+  .35 
+  .38 
+  .40 
+  .41 
+  .44 
+  .42 
+  .41 
+  .39 
+  .35 
+  .31 
+  .28 
+  .24 
+  .19 
+  .16 
+  .13 
+  .08 
+  .04 
+  .282 

Sept.  24 
lam 

2          

3 

[30.08] 

(30.26) 

4          
5 

6          

7 

8          

30.22 

29.82 

29.99 

30.22 

9          

10          
lla.m  

N 

[30.13] 

(20.80) 

1  p.m  
2 

3          
4          
5          

(29.80) 

[29.99] 

6          
7 

8          
9 

29.87 

29.93 

29.83 

29.91 

10          
11  p.m  
M 

(29'.  79) 

[29.96] 

The  insertion  of  the  six  maximum  and  minimum  points,  and  of  the 
corresponding  unbracketed  values  in  the  (3-6)  and  (4-5)  columns,  probably 
gave  a  slightly  increased  accuracy  to  the  computation  for  this  particular  day. 
These  maximum  and  minimum  points  can  be  determined  with  difficulty 
and  with  but  a  small  degree  of  accuracy  from  the  forecast  maps  by  studying 
the  rate  and  direction  of  progress  of  the  isobars  in  each  12-hour  interval. 
Usually,  the  maxima  and  minima  are  poorly  defined  and  differ  but  little 
from  the  values  which  would  be  obtained  by  direct  interpolation  between  the 
preceding  and  following  8  o'clock  values.  For  these  reasons  the  hourly 


36  EFFECTS   OF   WINDS   AND    OF 

barometric  effects  have  been  computed  as  a  rule  without  the  insertion  of 
maximum  and  minimum  points,  in  the  belief  that  the  possible  gain  in  ac- 
curacy thereby  sacrificed  is  negligible. 

The  separate  values  of  N-S  effect  and  of  E-W  effect,  and  their  sum  E\, 
the  total  barometric  effect,  were  computed  from  formula  (43).  Note  that 
the  computed  maximum  barometric  effect  occurred  at  noon  on  September  24 
corresponding  to  the  value  +40  in  the  column  headed  (3-6). 

To  obtain  hourly  elevations  of  the  mean  surface  of  Lake  Michigan- 
Huron  from  observed  hourly  elevations  of  the  water  surface  at  Milwaukee, 
one  must  subtract  the  barometric  effect  for  each  hour  computed  as  shown  in 
the  example. 

COMPUTATION   OF  DAILY  BAROMETRIC   EFFECTS. 

The  mean  of  the  24  hourly  barometric  effects  for  any  day  is  the  best 
value  obtainable  for  the  daily  barometric  effect. 

To  obtain  daily  elevations  of  the  mean  surface  of  Lake  Michigan-Huron, 
one  must  subtract  the  daily  barometric  effect  for  each  day  from  the  mean  of 
the  24  hourly  elevations  of  the  water  surface  at  Milwaukee  for  that  day. 

To  compute  the  barometric  effect  for  each  day  by  first  computing  the  24 
hourly  barometric  effects  for  that  day  and  then  taking  the  mean  is  an  un- 
necessarily slow  and  laborious  procedure.  The  method  of  computation 
indicated  below  is  much  more  rapid  and  is  believed  to  be  but  slightly  less 
accurate. 

In  the  formula  for  the  observation  equations  as  shown  in  (40),  page  24, 
the  first  eight  terms  &«,<>  BWQ,  bw\  Bw\  .  .  .  bnZ  BnZ,  bns  J5«j  express  the 
change  in  barometric  effect  from  one  day  to  the  next.  The  quantities 
Bwo,  Bwi,  .  .  .  BnZ,  EM  have  been  determined  by  the  least-square  solu- 
tions, and  their  values  are  shown,  for  each  of  five  stations,  in  table  No.  6, 
page  32.  Therefore,  the  following  formulae  express  the  change  in  the  baro- 
metric effect  from  one  day  to  the  next  at  the  stations  named: 
At  Buffalo 

-  2 . 22bw0 + 3 . 856*1 + 5 . 966«,2 + 1 . 856^  -  3 . 966n0 

-10.78&ni-11.69&n2-4.82&n3        (46) 
At  Cleveland 

+0.696^0+1 . 226wi+l .  046w2+l .  306w3+ 1 . 736n0 

+4 . 23&nl+6 . 046n2+ 1 . 606n3         (47) 
At  Milwaukee 

- 1 . 786«0  -  4 . 346^  -  2 . 926^  -  .  906w3 + 1  -  996n0 

+6 . 536ni+6 . 346n2+2 . 036,*        (48) 
At  Harbor  Beach 

+2. 816w0+5.296wl+5.106w2+. 686W3-1. 006n0 

-  . 846ni+ . 496n2+ . 47&«3        (49) 
At  Mackinaw 

-  . 916^-2. 05&«,2- 1 . 386w3+ . 386n0 

-  2 . 896nl  -  4 . 706n2  - 1 . 555n3         (50) 


BAROMETRIC   PRESSURES   ON   THE    GREAT  LAKES  37 

For  the  definitions  of  the  meanings  of  bw0,  bwi  .  .  .  6n2,  6n3,  consult  page 
24.  In  the  above  formulae  the  unit  in  which  these  quantities  are  to  be 
expressed  is  0.01  inch.  In  these  formulas  the  unit  in  which  the  barometric 
effect  is  expressed  is  in  units  of  0.001  foot. 

The  method  of  computation  actually  used  to  secure  the  barometric  effects 
for  each  day  consists  of  five  steps,  as  follows: 

(1)  The  hourly  barometric  effects  were  computed  by  the  method  set 
forth  on  pages  32-36  for  the  first,  the  last,  and  a  few  intermediate  days  of 
the  long  series  of  days  under  consideration.     The  daily  barometric  effect 
was  obtained  for  each  of  these  selected  days  by  taking  the  mean  of  the  24 
hourly  effects  for  that  day — the  most  accurate  method. 

(2)  The  change  in  barometric  effect  from  day  to  day  throughout  the 
whole  series  was  computed  from  the  appropriate  one  of  formulae  (46) 
to  (50). 

(3)  The  changes  from  day  to  day  computed  in  step  (2)  were  applied  one 
by  one  to  the  computed  barometric  effect  for  the  first  day  of  the  series  ob- 
tained in  step  (1)  to  secure  the  barometric  effect  for  each  day  in  turn  up  to 
and  including  the  next  selected  day  on  which  the  daily  barometric  effect  had 
been  computed  in  step  (1). 

(4)  At  this  point,  the  second  selected  day,  a  discrepancy  appeared  between 
the  barometric  effect  as  computed  for  this  day  by  step  (1)  and  as  computed 
by  step  (3).     This  discrepancy  was  distributed  proportionally  to  time  be- 
tween the  first  and  the  second  selected  day,  and  all  intermediate  values 
corrected,  so  as  to  make  the  discrepancy  disappear  at  the  second  selected 
day.     The  values  so  corrected  were  adopted  as  sufficiently  accurate. 

(5)  Steps  (3)  and  (4)  were  taken  from  the  second  to  the  third  selected  day, 
from  the  third  to  the  fourth  selected  day,  and  so  on  to  the  end  of  the  series. 

The  computation  for  August  26  to  September  24,  1910,  at  Milwaukee, 
made  in  accordance  with  the  above  statement,  is  shown  on  the  following 
page.  The  unit  is  one  foot. 

The  values  in  the  column  headed  "Change  in  barometric  correction" 
were  computed  from  formula  (48).  Each  value  is  the  change  in  barometric 
correction  from  the  day  before  to  the  day  indicated  on  the  line  on  which  the 
value  is  placed. 

Of  the  values  in  the  column  headed  "Barometric  correction  based  directly 
on  computed  values,"  those  included  in  parentheses  for  the  selected  days 
were  computed  from  hourly  barometric  effects  as  indicated  in  step  (1),  and 
the  others,  without  parentheses,  were  computed  as  indicated  in  step  (3) 
by  applying  from  day  to  day  the  changes  shown  in  the  preceding  column. 

At  the  second  selected  day,  September  4,  there  are  two  values  in  the  third 
column,  namely,  +0.045,  computed  by  step  (3),  and  (  —  0.070),  computed 
by  step  (1).  The  discrepancy  is  —0.115  foot.  This  discrepancy  was  dis- 
tributed over  the  interval  August  26-September  4  at  — =0.0128  foot 

per  day,  as  shown  in  the  fourth  column. 


435704 


38  EFFECTS   OF   WINDS   AND    OF 

Example  of  computation  of  daily  barometric  effects  (Milwaukee,  Aug.  2(i  to  Sept.  %4,  1910). 


Date. 

Barometric  correction. 

Elevation  of 
water  surface. 

Change  in 
correction. 

Based  directly 
on  computed 
values. 

To  conform 
to  values 
computed 
by  step  (1). 

Final. 

Observed. 

Corrected. 

Aug.  26... 

(+.238) 

.000 

+  .238 

580.00 

580.238 

27... 

-.ois 

+  .220 

-.013 

+  .207 

.14 

.347 

28... 

-.176 

+  .044 

-.026 

+  .018 

.33 

.348 

29... 

-.098 

-.054 

-.038 

-.092 

.34 

.248 

30... 

+  .058 

+  .004 

-.051 

-.047 

.34 

.293 

31... 

+  .059 

+  .063 

-.064 

-.001 

.24 

.239 

Sept.    I... 

-.015 

+  .048 

-.077 

-.029 

.32 

.291 

2... 

+  .011 

+  .059 

-.090 

-.031 

3... 

+  .065 

+  .124 

-.102 

+  .022 

4... 

-.079 

+  .045  (-.070) 

.000 

-.070 

5... 

-.096 

-.166 

.000 

-.166 

580  !  25 

580  !  084 

6... 

+  .232 

+  .066 

+  .001 

+  .067 

.07 

.137 

7  ... 

+  .019 

+  .085 

+  .002 

+  .087 

.00 

.087 

8... 

-.022 

+  .063 

+  .002 

+  .065 

.23 

.295 

9... 

+  .061 

+  .124 

+  .002 

+  .126 

.25 

.376 

10... 

-.046 

+  .078 

+  .003 

+  .081 

.25 

.331 

11... 

-.112 

-.034 

+  .004 

-.030 

.29 

.260 

12... 

-.077 

-.111 

+  .004 

-.107 

.51 

.403 

13... 

+  .093 

-.018 

+  .004 

-.014 

.37 

.356 

14... 

+  .105 

+  .087  (+.092) 

.000 

+  .092 

.25 

.342 

15... 

-.022 

+  .070 

.000 

+  .070 

.31 

.380 

16... 

-.019 

+  .051 

+  .001 

+  .052 

.30 

.352 

17... 

+  .001 

+  .052 

+  .001 

+  .053 

.16 

.213 

18... 

-.198 

-.146 

+  .002 

-.144 

.44 

.296 

19... 

+  .018 

-.128 

+  .002 

-.126 

.40 

.274 

20... 

+  .162 

+  .034 

+  .002 

+  .036 

.23 

.266 

21... 

-.090 

-.056 

+  .003 

-.053 

.34 

.287 

22... 

-.043 

-.099 

+  .003 

-.096 

.40 

.304 

23... 

-.062 

-.161 

+  .004 

-.157 

.51 

.353 

24... 

-.125 

-.286  (-.282) 

.000 

-.282 

.71 

.428 

The  corrections  shown  in  the  fourth  column  were  applied  to  the  values 
shown  in  the  third  column  to  obtain  the  final  adopted  values  of  the  baro- 
metric corrections  as  shown  in  the  fifth  column. 

The  rule  ordinarily  observed  in  regard  to  selected  days  was  to  place  them 
not  more  than  1  month  apart  in  any  case  and  not  more  than  10  days  apart 
over  any  interval  through  which  the  discrepancy  to  be  distributed  exceeded 
0.070  foot. 

The  discrepancy  in  question  is  due  to  two  causes:  (a)  to  omitted  decimal 
places  in  the  computations,  and  (6)  to  the  fact  that  the  process  of  computing 


BAROMETRIC   PRESSURES   ON   THE    GREAT   LAKES  39 

the  hourly  barometric  effects,  and  thence  the  daily  barometric  effects,  by 
step  (1)  involves  a  smoothing  out  of  the  discrepancies  between  the  values  of 
Bwa,  Bwi,  .  .  .  Bn-i,  Bns  as  derived  from  the  least-square  solution,  and 
therefore  does  not  agree  exactly  with  the  computations  made  from  formulae 
(46)  to  (50).  The  rule  stated  in  the  preceding  paragraph  was  adopted,  as  a 
result  of  experience,  as  probably  giving  the  best  balance  between  extreme 
accuracy,  on  the  one  hand,  and  large  expenditure  of  time  in  computation, 
on  the  other  hand. 


THEORETICAL  BASIS  FOR  WIND  OBSERVATION  EQUATIONS. 

The  formula  for  wind  effect  which  has  been  adopted  as  the  basis  for  the 
observation  equations  is  as  follows: 


;*Qs*  (51> 


W  is  the  effect,  at  a  given  time,  of  the  wind,  at  a  given  gage  station,  in 
elevating  the  water  surface  at  that  station  above  the  mean  elevation  of  the 
whole  surface  of  the  lake,  h  is  the  velocity  of  the  wind  at  that  station. 
Sz  is  a  quantity  appropriate  to  the  station,  for  each  wind  direction,  which 
expresses  the  relation  between  the  effect  of  a  wind  of  a  certain  velocity,  on 
the  one  hand,  and  the  depth  of  the  lake  at  every  point,  the  shape  of  the 
bottom,  the  horizontal  dimensions  of  the  lake,  and  the  shape  of  its  shore,  on 
the  other  hand. 

A  wind  blowing  across  a  lake  drives  the  surface  water  to  leeward  at  every 
point  on  the  surface  at  a  rate  dependent  upon  the  wind  velocity.  This 
surface  water  delivered  toward  the  lee  shore  tends  to  raise  the  surface 
elevation  of  that  part  of  the  lake.  As  soon  as  this  action  has  established  a 
surface  slope  downward  to  windward,  gravity  tends  to  set  up  a  current  to 
windward,  which  current  extends  to  the  full  depth  of  the  lake,  from  the  sur- 
face to  the  bottom.  Said  return  current  to  windward  is  a  function  of  the 
surface  slope,  tending  to  be  greater  the  greater  the  slope.  When  a  steady 
regime  has  been  established  for  a  wind  of  a  certain  direction  and  velocity, 
the  total  volume  of  water  per  unit  of  time  delivered  to  windward  across  any 
line  which  may  be  drawn  completely  across  the  lake  is  necessarily  equal  to 
the  total  volume  of  surface  water  delivered  per  unit  of  time  to  leeward 
across  that  line.  If  it  were  otherwise,  the  surface  elevations  at  some  parts 
of  the  lake  would  necessarily  be  changing  and  the  regime  would  not  be  steady. 

Formula  (51)  expresses  the  wind  effect  after  the  steady  regime  has  been 
established  for  a  wind  of  any  given  direction  and  velocity. 

The  explanation  of  the  theoretical  basis  of  formula  (51)  is  given  in  the 
following  successive  steps: 

(1)  For  an  assumed  wind  of  constant  direction  and  constant  velocity,  and 
for  a  narrow  strip  of  the  lake  parallel  to  the  direction  of  the  wind,  the 
relation  at  each  point  of  the  strip,  during  a  steady  regime,  between  the 


40  EFFECTS   OF   WINDS   AND   OF 

depth  of  the  water  and  its  surface  slope  is  established.  In  this  part  of  the 
exposition,  each  of  the  various  strips,  each  parallel  to  the  wind  direction, 
across  the  lake  from  windward  to  leeward  is  assumed  to  act  independently 
of  every  other  such  strip. 

(2)  It  is  recognized  that  the  various  strips  across  a  lake  do  not  act  in- 
dependently.    The  method  of  applying  the  relation  derived  in  (1)  to  an 
actual  lake  with  its  irregular  bottom  and  shores  is  set  forth. 

(3)  A  statement  is  made  in  regard  to  the  manner  in  which  the  exponent 
of  h  (2.4)  has  been  derived. 

RELATION  BETWEEN  DEPTH  AND  SLOPE  PRODUCED  BY 

WIND. 

The  Chezy  formula  for  the  flow  of  water  in  an  open  channel  is  the  standard 
formula  usually  given  in  the  text-books  on  hydraulics.  It  is  the  fundamental 
basis  from  which  other  more  elaborate  formulae  have  been  built  up. 

The  Chezy  formula  is 

V  =  C(RS)*  (52) 

In  this  formula, 

V  =  the  mean  velocity  of  flow  in  a  cross-section  of  the  channel. 

C  =  an  empirical  coefficient  depending  in  the  main  upon  the  roughness 

of  the  solid  surfaces  of  the  channel,  and  also  upon  the  velocity 

of  flow,  upon  the  hydraulic  radius,  and  possibly  upon  the 

slope  of  the  water  surface. 
12  =  the  mean  hydraulic  radius  =  the  area  of  the  cross-section  of  the 

stream  of  water  divided  by  its  wetted  perimeter. 
/S  =  the  slope  of  the  water  surface. 

This  Chezy  formula  is  adopted  as  a  part  of  the  basis  for  the  following 
derivation  of  the  relation  between  the  wind  and  the  slope  of  the  water  surface 
ultimately  produced  by  it. 

Consider  the  conditions  on  a  lake  during  a  period  when  a  wind  of  uniform 
constant  velocity  and  direction  is  blowing  over  the  surface  of  the  lake. 
Consider  any  strip  of  surface  of  the  lake,  of  unit  width,  with  the  axis  of  the 
strip  parallel  to  the  direction  of  the  wind.  So  long  as  the  wind  remains 
constant  in  velocity  and  direction  it  is  evident  that  the  rate  at  which  the 
surface  water  will  be  delivered  to  leeward  along  the  strip  by  the  action  of 
the  wind  will  be  approximately  constant.  Let  the  volume  of  water  delivered 
per  unit  of  time  past  any  line  across  the  strip  under  the  action  of  a  wind  of 
fixed  velocity  be  called  Q. 

The  velocity  of  this  surface  drift  caused  by  the  wind  is  evidently  a  maxi- 
mum at  the  surface  where  the  wind  acts  upon  the  water,  diminishes  gradually 
with  increase  of  depth  below  the  surface,  and  becomes  zero,  or  practically 
so,  at  a  moderate  depth  below  the  surface.  This  is  illustrated  by  figure  3, 
plate  4,  in  which  the  arrows  are  proportional  to  the  velocities  under  con- 


BAROMETRIC   PRESSURES   ON   THE    GREAT   LAKES  41 

sideration.  The  exact  relation  between  these  various  arrows  is  not  claimed 
to  be  known,  and  for  the  purpose  of  this  investigation  it  is  not  necessary  to 
know  it.  The  illustration  figure  3  is  meant  merely  to  show  the  general 
conception. 

As  the  wind  delivers  water  toward  the  lee  shore,  as  an  east  wind  would 
deliver  water  along  the  line  BB,  figure  1,  plate  4,  on  Lake  Erie  toward  the 
western  shore  of  the  lake,  it  raises  the  elevation  of  the  water  surface  on  that 
leeward  portion  of  the  lake.  In  due  time,  if  the  wind  remains  constant  in 
velocity  and  direction,  a  surface  slope  will  be  established  everywhere  on  the 
strip,  along  BB,  downward  to  windward  such  that  the  return  current  to 
windward,  set  up  by  the  action  of  gravity,  will  deliver  the  water  to  windward 
at  the  same  rate,  Q,  that  the  wind  delivers  it  to  leeward.  During  the  con- 
tinuance of  this  steady  regime,  after  it  is  once  established,  the  elevation  of 
the  water  surface  at  every  part  of  the  strip  will  remain  fixed,  the  volume  of 
water  delivered  by  the  wind  past  each  point  of  the  strip  per  unit  of  time 
to  leeward  will  be  Q,  and  the  volume  delivered  per  unit  of  time  past  any 
point  by  the  return  gravity  current  to  windward  will  also  be  Q. 

The  return  current  is  illustrated  by  figure  4,  plate  4.  As  this  current  is 
produced  by  gravity,  the  Chezy  formula  (52)  expresses  the  relation  neces- 
sarily existing  between  the  velocity  and  the  slope.  This  return  current, 
being  produced  by  gravity,  extends  throughout  the  depth,  and  with  suf- 
ficient accuracy  for  the  present  purpose  may  be  considered  as  being  the  same 
at  all  depths  as  shown  in  the  illustration. 

As  the  wind  current  indicated  in  figure  3,  plate  4,  to  leeward  and  the 
return  gravity  current  to  windward,  figure  4,  exist  at  the  same  time,  the 
actual  velocities  of  the  water  at  various  depths  are  properly  represented  by 
figure  5,  in  which  each  arrow  is  the  algebraic  sum  of  the  two  corresponding 
arrows  in  figures  3  and  4.  During  the  steady  regime  under  consideration 
the  sum  of  the  arrows  in  figure  3  is  the  same  as  the  sum  of  the  arrows  in 
figure  4,  and  in  figure  5  the  sum  of  the  arrows  which  point  to  the  left  is  the 
same  as  the  sum  of  the  arrows  which  point  to  the  right.  According  to  these 
figures,  the  net  delivery  of  water  past  the  point  is  zero,  the  separate  volumes 
delivered  per  unit  of  time  in  opposite  directions  each  being  Q. 

As  the  Chezy  formula  (52)  holds  for  the  return  current  produced  by 
gravity,  the  following  relations  (53)  to  (59)  are  true  during  the  continuance 
of  the  steady  regime: 

At  each  point  of  the  strip  of  unit  width  under  consideration, 

R  =  D  =  depth  of  water  at  that  point  of  the  strip.  (53) 

This  follows  from  the  fact  that  the  only  part  of  the  perimeter  of  the  cross- 
section  of  the  stream  under  consideration  which  encounters  resistance  at  a 
solid  boundary  is  the  bottom.  The  two  sides  of  the  stream  are  in  contact 
with  water  of  adjacent  strips.  The  strip  being  of  unit  width  and  the  wetted 
perimeter  unity,  the  area  of  the  cross-section  of  the  stream  flowing  along  the 
strip  is  numerically  the  same  as  the  depth. 


42  EFFECTS   OF   WINDS   AND   OF 

Under  the  conditions  stated, 

V-  _  ?  _  -«  (54) 

area  of  cross-section  of  the  stream      D 

By  substitution  from  (53)  and  (54)  in  (52)  there  is  obtained 


From  the  foregoing  equation  by  solution  for  S  there  is  obtained 


in  which  —  is  a  constant,  during  the  steady  regime  under  consideration, 
C 

which  will  be  called  Ci. 

Qt 

This  constant,  C\=—^  'depends  on  the  influences  which  fix  C,  the  Chezy 

coefficient,  and  upon  the  action  of  the  wind  upon  the  water,  which  fixes  Q. 
Equation  (55)  may  for  the  present  purpose  be  conveniently  written  thus  : 


(i) 


Let  the  difference  of  elevation  of  the  surface  of  the  water  at  any  two  points 
along  the  axis  of  the  strip  under  consideration  be  called  Hd  and  let  the  dis- 
tance between  the  two  points  be  called  L. 

Then 

Hd  =  SL  (57) 

From  (56)  and  (57),  for  any  short  portion  of  the  strip  on  which  D  is  con- 
stant, 

(58) 


In  general,  under  the  influence  of  a  steady  wind,  the  surface  of  the  water 
along  any  strip  during  the  steady  regime  will  have  some  such  shape  as  that 
shown  in  the  curve  labeled  "disturbed  water  surface"  in  figure  1,  plate  4. 
On  that  curve,  which  is  drawn  for  a  strip  on  Lake  Erie  along  the  line  BB 
and  for  a  west  wind,  the  surface  of  the  western  part  of  the  strip  is  shown 
depressed  below,  and  along  the  eastern  part  elevated  above,  the  normal 
elevation  which  it  would  have  if  no  wind  were  blowing.  One  point  on  the 
curve,  there  shown  at  1,027  thousands  of  feet  west  of  the  Buffalo  gage,  is 
shown  as  unchanged  in  elevation  by  the  wind.  Call  such  a  point  the  nodal 
point  of  the  strip. 

Let  H  be  the  total  disturbance  of  elevation  of  any  point  on  the  strip  under 
consideration. 


BAROMETRIC   PRESSURES   ON   THE   GREAT   LAKES  43 

Then,  from  (58),  it  is  evident,  if  one  thinks  of  a  step-by-step  integration  of 
differences  of  elevation  from  the  nodal  point  of  the  strip  to  the  point  under 
consideration,  that  T 

H  =  C^±  (59) 

The  summation,  indicated  by  S  in  this  formula,  of  terms  each  of  the  form 
—  is  supposed  to  be  made  in  portions  of  length  L  so  short  that  along  each 

portion  D  (the  depth),  may  be  considered  constant.  The  summation  is 
supposed  to  extend  over  the  whole  distance  from  the  nodal  point  to  the  point 
under  consideration. 

Note  that  if  the  point  under  consideration  is  at  a  gage,  H  is  the  disturbance 
of  the  elevation  of  the  water  surface  at  the  gage  which  is  produced  by  the 
wind. 

DERIVATION   OF  Sx. 

In  what  has  preceded,  culminating  in  formula  (59),  only  a  single  strip,  of 
unit  width,  parallel  to  the  direction  of  the  wind  has  been  considered. 

It  is  evident,  if  one  reviews  the  various  steps  by  which  (59)  was  derived, 
that  it  is  true  for  strips  of  any  one  width  within  which  there  is  no  transverse 
variation  of  depth.  The  slope  of  the  water  surface  along  a  strip  is  inde- 
pendent of  the  width  of  the  strip  provided  the  depth  of  water  is  the  same  for 
the  whole  width  of  the  strip. 

Consider  an  actual  lake  to  be  divided  up  into  strips  of  moderate  width, 
each  parallel  to  the  wind  direction.  Equation  (59)  applied  to  each  strip 
independently  would  show  the  values  of  H  differing  at  adjacent  points  on 
adjacent  strips.  Clearly,  if  such  a  condition  actually  existed  for  a  moment 
the  water  would  begin  to  flow  across  the  arbitrarily  assumed  (but  non- 
existent) boundary  between  strips,  from  the  higher  of  the  two  points  to  the 
lower.  Such  cross-currents  would  tend  always  to  bring  adjacent  parts  of 
the  various  strips  to  one  elevation,  and  thereby  tend  to  modify  somewhat 
the  slope  along  each  strip. 

In  any  actual  lake,  with  its  irregular  depths  and  irregular  shore  line,  there 
will  certainly  be  such  cross-currents.  A  study  of  the  complicated  relations 
involved  leads  to  the  conclusion  that  such  cross-currents  are  sluggish  in 
general  except  in  shallow  water  near  shore,  and  that  assumption  No.  4, 
stated  below,  is  near  the  truth  and  leads  to  errors  so  small  as  to  be  negligible 
in  comparison  with  other  errors  of  this  investigation  which  are  unavoidable. 

ASSUMPTION  No.  4. 

It  is  assumed  that  the  return  gravity  current  has  small  components  at 
right  angles  to  the  wind  direction  that  cross  the  boundaries  between 
strips  in  such  a  manner  as  to  bring  adjacent  portions  of  the  various  strips 
nearly  to  the  same  elevation  and  that  the  slope  on  each  strip  thereby  estab- 
lished is  the  same  as  if  the  depth  everywhere  along  each  strip  were  the 
same  as  the  mean  depth  for  all  abreast  portions  of  all  strips.  For  example, 


44  EFFECTS   OF   WINDS  AND   OF 

if  the  strip  under  consideration  lay  along  the  line  BB  in  figure  1,  plate  4,  on 
the  actual  Lake  Erie,  the  slope  at  point  F  on  the  strip  is  that  fixed  by 
formula  (56)  if  one  uses  for  D  not  the  depth  at  F  but  the  mean  depth 
along  the  line  CC',  which  is  at  right  angles  to  BB  through  the  point  F. 

Under  assumption  No.  4,  formula  (59)  applies  to  any  strip  parallel  to  the 
wind  direction  on  any  actual  lake  provided  one  uses  for  each  point  on  the 
strip  a  depth  D  which  is  a  mean  depth  along  such  a  line  as  CC'  at  right 
angles  to  the  strip  through  the  point. 

The  values  of  2,,  the  summation  indicated  in  (59),  have  been  computed 
for  8  wind  directions  for  the  Buffalo  and  Cleveland  gages  on  Lake  Erie  and 
for  the  Milwaukee,  Harbor  Beach,  and  Mackinaw  gages  on  Lake  Michigan- 
Huron.  The  way  in  which  formula  (59)  and  assumption  No.  4  have  been 
applied  in  these  computations  will  be  illustrated  by  selected  portions  of  the 
computations. 

EXAMPLE  OF  COMPUTATION  OF  s*. 

The  whole  area  of  Lake  Erie,  as  shown  on  a  chart  of  Lake  Erie  issued  by 
the  United  States  Lake  Survey,  was  divided  up  into  strips  parallel  to  CC' 
on  figure  1,  plate  4,  the  axis  of  each  strip  being  in  the  meridian.  The  widths 
of  the  strips  varied  from  2,000  feet  to  50,000  feet.  The  strips  were  wide 
where  the  depths  were  large  and  comparatively  regular,  and  were  narrow  in 
shallow  water  and  where  the  depths  were  irregular.  The  division  lines 
having  been  drawn  on  the  chart,  the  separate  depths  were  estimated  from 
the  chart  and  entered  in  the  computation  illustrated  by  selected  portions  in 
table  No.  8. 

In  table  No.  8  the  strip  limits  are  given,  in  the  first  column,  in  thousands 
of  feet  measured  westward  from  the  Buffalo  gage.  The  first  strip  shown  in 
the  table  has  for  its  eastern  limit  a  meridian  line  which  is  50,000  feet  west 
of  the  Buffalo  gage,  and  for  its  western  limit  one  which  is  100,000  feet  west 
of  the  Buffalo  gage.  The  approximate  location  of  each  of  the  strips  may  be 
seen  on  figure  1  of  plate  4,  on  which  a  scale  of  distances  westward  from  the 
Buffalo  gage  is  shown. 

With  the  chart  before  one,  and  with  dividers  in  hand,  it  is  a  matter  of 
easy  routine  to  visually  divide  any  strip  across  the  lake  into  ten  equal  parts, 
to  estimate  the  mean  depth  in  each  part,  and  to  enter  it  in  the  computation 
as  shown  in  the  second  to  the  eleventh  columns  of  table  No.  8.  The  mean 
depth  shown  on  each  line  in  the  twelfth  column  is  the  mean  of  the  depths 
entered  in  the  next  preceding  ten  columns  on  that  line. 

The  values  of  D3  as  shown  in  table  No.  8  are  rounded  off  to  three  sig- 
nificant figures.  That  gives  sufficient  accuracy. 

Each  L  is  a  distance  parallel  to  the  wind  under  consideration — an  east 
wind  in  this  case.  It  is  in  each  case  the  same  as  the  width  of  the  strip,  which 
is  the  difference  between  the  two  figures  given  in  the  first  column  as  strip 
limits.  For  convenience,  L  is  given  in  thousands  of  feet  in  the  table. 

The  first  group  of  strips  shown  in  table  No.  8  includes  the  deepest  part  of 


BAROMETRIC   PRESSURES   ON   THE    GREAT   LAKES 


45 


Lake  Erie,  southeast  of  Long  Point.     The  depths  are  comparatively  regular 
and  the  strips  are  50,000  feet  (nearly  10  miles)  wide.     Note  the  relatively 

small  values  of  — ,  even  though  values  of  L  are  large. 

The  second  group  of  strips  shown  in  table  No.  8  includes  that  part  of  the 


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(N  (N  (N  (N  <N 

46  EFFECTS   OF  WINDS   AND    OF 

lake  which  is  near  Cleveland.  In  this  region  the  bottom  of  the  lake  is 
nearly  level  over  about  nine-tenths  of  the  width  of  the  lake,  the  depth  being 
from  10  to  14  fathoms  except  near  each  shore.  The  shape  of  the  cross- 
section  of  the  lake  in  this  region  is  illustrated  by  figure  2,  plate  4,  which  is 
drawn  to  scale.  Two  of  the  strips  are  50,000  feet  wide.  The  other  three 
strips  were  made  narrower  simply  for  the  purpose  of  making  one  strip 
boundary  run  through  the  Cleveland  gage,  which  is  758,000  feet  west  of  the 
Buffalo  gage. 

The  third  group  of  strips  shown  in  table  No.  8  lies  in  the  very  shallow 
portions  of  the  lake  near  its  western  end.  The  meridian,  which  is  1,200,000 
feet  west  of  Buffalo  gage,  passes  between  the  Toledo  harbor  light,  near  the 
outer  end  of  the  dredged  channel  to  Toledo,  and  Cedar  Point  on  the  south 
shore  of  the  lake.  The  extreme  western  end  of  the  lake  is  1,239,000  feet 
west  of  Buffalo  gage.  Note  that  in  this  group  of  strips  the  depths  vary 
from  3  fathoms  to  less  than  one-half  of  a  fathom  (which  is  shown  as  zero  in 
the  table).  Note  that  L  is  small  in  this  group,  only  5,000  feet,  and  that 

nevertheless  the  values  of  —  are  relatively  large,  — 14.6  for  the  last  strip 

shown. 

Sandusky  Bay  was  covered  by  a  computation,  similar  to  that  illustrated 
by  table  No.  8,  separately  from  the  main  computation  which  covered  the 
lake.  This  was  the  only  case  so  treated  for  east  or  west  winds  across  Lake 
Erie.  In  general,  it  was  found  necessary  to  make  such  separate  computa- 
tions for  bays  or  portions  of  bays  which  are  not  covered  by  continuous 
strips  across  the  lake  transverse  to  the  assumed  wind,  the  bay  being  in  each 
such  case  cut  off  from  the  lake  on  each  strip  by  an  intervening  point  of  land. 
In  some  of  the  computations  made  for  certain  directions  of  wind,  and  es- 
pecially in  Lake  Michigan-Huron,  there  were  several  such  special  auxiliary 
computations  for  bays. 

When  such  a  computation  as  that  illustrated  by  table  No.  8  had  been 
completed  for  a  given  assumed  wind  and  lake,  a  computation  such  as  that 
illustrated  in  table  No.  9  was  made. 

The  strip  limits  are  given  in  the  first  column  of  table  No.  9  in  thousands 
of  feet  measured  westward  from  the  Buffalo  gage,  just  as  they  were  given  in 
table  No.  8. 

Each  value  of  L  corresponding  to  the  width  of  a  strip  is  given  in  thousands 
of  feet. 

The  third  column  of  table  No.  9  gives  the  length  of  each  strip  measured  at 
right  angles  to  the  assumed  wind  in  thousands  of  feet.  The  entry  110  in 
the  third  column  for  the  strip  50  to  100  in  table  No.  9  means  that  said 
strip  is  110,000  feet  long  from  the  Ontario  (Canadian)  shore  to  the  Ohio 
shore. 

The  product  of  the  width  of  any  strip  as  shown  in  the  second  column  by 
the  length  of  that  strip  as  shown  in  the  third  column  is  its  area  as  shown  in 
the  fourth  column,  expressed  in  units  of  1,000,000  square  feet. 


BAROMETRIC   PRESSURES   ON   THE    GREAT   LAKES 


47 


TABLE  No.  9. — Computation  of  2 j  for  E-W  axis,  Lake  Erie. 


Strip  limits. 

L 

Length 
of  strip. 

Area 
of  strip. 

4 

to  far  side 
of  strip. 

4 

to  middle 
of  strip 
=  (*>). 

(*>) 
times 
area. 

Continuous 
sum  of 
preceding 
column. 

50  to     100 
100          150 
150         200 
200         250 
250         300 

50 
50 
50 
50 
50 

110 
144 
177 
207 
220 

5,500 
7,200 
8,850 
10,350 
11,000 

160 
160 
161 
161 
161 

160 
160 
161 
161 
161 

880,000 
1,150,000 
1,425,000 
1,670,000 
1,770,000 

1,450,000 
2,600,000 
4,030,000 
5,700,000 
7,470,000 

650         700 
700         750 
750          755 
755          758 
753         800 

50 
50 
5 
3 
42 

332 
346 
342 
340 
319 

16,600 
17,300 
1,710 
1,020 
13,400 

162 
162 
162 
162 
162 

162 
162 
162 
162 
162 

2,690,000 
2,800,000 
277,000 
165,000 
2,170,000 

24,900,000 
27,700,000 
28,000,000 
28,200,000 
30,300,000 

1,200       1,205 
1,205       1,210 
1,210       1,215 
1,215       1,220 
1,220       1,225 

5 
5 
5 
5 
5 

66 
68 
66 
56 

48 

330 
340 
330 
280 
240 

185 
190 
196 
206 
221 

183 

188 
193 
201 
113 

60,400 
63,900 
63,700 
56,300 
27,100 

44,400,000 
44,400,000 
44,500,000 
44,500,000 
44,600,000 

In  the  fifth  column  there  is  shown  the  continuous  summation  of  —  step 
by  step  from  the  extreme  eastern  part  of  Lake  Erie,  near  Buffalo,  to  the 
far  side  of  the  strip.  The  values  of  S —  shown  in  this  fifth  column  were 

obtained  by  adding  successively  the  values  of  —  computed  as  shown  in 
table  No.  8.  For  example,  S —  for  the  west  (or  far)  side  of  strip  1,200  to 

1,205  is  shown  as  185  in  the  fifth  column  of  table  No.  9.  The  value  of  — 
for  strip  1,205  to  1,210  is  shown  as  5.00  in  the  last  column  of  table  No.  8. 
In  the  fifth  column  of  table  No.  9  the  value  of  S —  for  the  west  (or  far)  side 

of  strip  1,205  to  1,210  is  shown  as  190  (which  is  185+5.00). 

Each  value  in  the  sixth  column  of  table  No.  9,  called  (6)  for  convenience, 

is  the  mean  of  the  values  of  2 —  at  the  two  sides  of  the  strip  as  shown  in  the 
fifth  column. 


48  EFFECTS   OF   WINDS   AND   OF 

Each  value  shown  in  the  seventh  column  is  the  product  of  the  two  quanti- 
ties in  that  line  entered  in  the  fourth  and  sixth  columns.  Each  value  in  the 
seventh  column  is  rounded  off  to  three  significant  figures,  as  that  gives 
sufficient  accuracy. 

The  continuous  sum  shown  in  the  last  column  of  table  No.  9  was  started 
at  the  extreme  eastern  part  of  Lake  Erie  and  carried  forward  continuously, 
and  was  rounded  off  to  three  significant  figures  after  the  summation  had  been 
made  continuously. 

DETERMINATION  OF  POSITION  OF   NODAL  LINE. 

The  values  of  S—  shown  in  the  fifth  and  sixth  columns  of  table  No.  9 

are  all  referred  to  the  extreme  eastern  part  of  Lake  Erie,  near  Buffalo.  For 
use  in  formula  (59)  it  is  necessary  to  have  these  summations  referred  to  the 
nodal  point  on  the  line  of  disturbed  elevations  of  the  water  surface,  such 
as  is  indicated  by  the  point  G  in  figure  1,  plate  4.  Hence,  the  location  of 
this  point  G  must  be  determined. 

Figure  1,  plate  4,  is  drawn  for  a  west  wind.  It  shows  the  disturbed  water 
surface  depressed  to  the  westward  of  G  and  elevated  to  the  eastward.  It  is 
clear  that  if  the  wind  were  from  the  east  the  disturbed  water  surface  would 
be  depressed  to  the  eastward  of  the  nodal  point,  G,  and  elevated  to  the  west- 
ward. The  ultimate  effect  of  a  steady  east  wind  is  to  transfer  water  from 
portions  of  the  lake  lying  to  the  eastward  of  the  nodal  point  to  portions  of 
the  lake  lying  to  the  westward  of  that  point. 

In  the  computation  of  which  selected  parts  are  shown  in  table  No.  9  the 
summations  shown  in  the  fifth,  sixth,  and  eighth  columns  are  all  referred  to 
an  initial  point  at  the  extreme  eastern  part  of  the  lake  at  the  water  surface. 
This  point  is  the  lowest  point  of  the  disturbed  water  surface  under  the  in- 
fluence of  an  east  wind.  Let  the  total  disturbance  of  elevation  of  the  water 
by  an  east  wind  at  this  particular  point,  which  has  been  used  as  the  initial 
point,  be  called  Hi. 

The  total  amount  of  water  in  the  lake  which  is  above  the  elevation  of  this 
initial  point  in  the  disturbed  condition  under  the  influence  of  an  east  wind  is 


S  (area  of  strip)  (CiS —)  (60) 

In  (60)  the  outer  S  stands  for  a  summation,  strip  by  strip  for  the  strips 
shown  in  the  computations  illustrated  in  tables  Nos.  8  and  9,  of  the  products 

of  the  area  of  each  strip  by  the  quantity  Ci?—  for  that  strip.  This 
summation  is  supposed  to  start  at  the  initial  point  and  to  extend  over  the 
whole  lake.  So,  too,  the  summation  S —  for  each  strip  is  supposed  to 
start  at  the  same  initial  point,  but  is  supposed  to  stop  for  each  strip  at  that 


BAROMETRIC   PRESSURES   ON   THE    GREAT   LAKES  49 

strip.     From  (58)  it  is  clear  that  the  quantity  Ci2  —  at  each  strip  is  the 

mean  elevation  of  the  water  surface  of  that  strip  (in  the  disturbed  condition) 
referred  to  the  initial  point  as  a  zero.  Hence,  the  product  of  the  area  of  the 

strip  into  CiS—  for  that  strip  is  the  amount  of  water  in  that  strip  which 

lies  above  the  elevation  of  the  initial  point,  and  it  is  clear  that  the  grand 
summation  indicated  in  (60)  is  the  total  amount  of  water  in  the  lake  which 
is  above  the  initial  point  during  the  disturbed  condition. 

When  the  surface  of  the  lake  is  undisturbed  its  whole  surface  is  at  one 
elevation,  and  the  total  amount  of  water  in  the  lake  then  lying  above  the 
elevation  of  the  initial  point  under  consideration  is 

(Hi)  (area  of  lake)  (61) 

A  wind  blowing  the  surface  of  the  lake  does  not  change  the  total  content  of 
the  lake.  Hence,  the  total  amount  of  water  above  the  initial  point  (which 
was  the  lowest  point  of  the  disturbed  surface)  must  be  the  same  in  the  dis- 
turbed and  the  undisturbed  condition.  Hence,  (60)  and  (61)  are  equal. 
By  placing  them  so  in  an  equation  and  solving  for  Hi  there  is  obtained 


S  (area  of  strip)   CiS-gr      CiS  (area  of  strip)  z  ( 

£21  r=r  -  =  - 

area  of  lake  area  of  lake 

Equation  (62)  is  an  expression  for  the  disturbance  of  elevation  of  the  water 
surface  at  the  initial  point  referred  to  the  nodal  point,  of  which  the  location 
is  as  yet  unknown,  d  may  be  taken  outside  both  summation  signs  as  shown 
because  it  is  a  constant  which  is  the  same  for  all  strips. 

Equation  (59)  is  an  expression  for  the  disturbance  of  elevation  of  any  point 
on  the  lake.  Hence,  applying  this  formula  (59)  to  the  initial  point  under 
consideration,  there  is  obtained 

Hi  =  C&—  (63) 

in  which  the  summation  S  —  extends  from  the  nodal  point  to  the  initial 

point. 

By  placing  the  two  expressions  for  Hi  of  (62)  and  (63)  equal  to  each  other 
and  dividing  each  side  of  the  equation  by  Ci  there  is  obtained,  as  an  equation 
applicable  at  the  nodal  point  only, 


L      S  (area  of  strip)  ( S  —  ) 
'-^  =  V   D3/ 


area  of  lake 


50  EFFECTS   OF   WINDS   AND    OF 

Equation  (64)  identifies  the  value  of  S —  as  shown  in  the  fifth  and  sixth 

columns  of  the  computation  illustrated  in  table  No.  9,  at  which  the  nodal 
point  lies.  In  table  No.  9  the  seventh  column  shows  the  value  for  each 
strip  of  one  of  the  products  indicated  in  the  numerators  of  the  second 
member  of  (64).  The  continuous  summation  of  these  products  is  shown  in 
the  eighth  column.  This  summation  (eighth  column)  at  the  end  of  the 
computation,  covering  the  whole  lake,  is  the  numerator  of  the  second 
member  of  (64).  The  denominator  of  the  second  member  of  (64),  the  area 
of  the  lake,  is  the  sum  of  the  separate  areas  of  the  strips  as  shown  in  the 
fourth  column  of  table  No.  9. 

In  the  concrete  case  illustrated  in  table  No.  9  the  value  of  the  numerator  of 
the  second  member  of  (64)  was  found  to  be  45,500,000.  The  area  of  the 
lake  was  found  from  the  computation  to  be  276,000.  Hence,  the  value  of 

L  c        •,  .     ,     45,500,000     1C_      _>     .          ,. 

S —  at  the  nodal  point  was  found  to  be  — = =  165.     By  inspecting 

jD8  276,000 

the  fifth  and  sixth  columns  of  the  computation  illustrated  in  table  No.  9  it 
was  found  that  this  value  of  S —  occurred  at  1,027,000  feet  west  of  the 

Buffalo  gage  and  that  therefore  the  nodal  point  and  nodal  line  are  in  that 
location,  as  indicated  in  figure  1,  plate  4. 

The  preceding  derivation  of  certain  formulae  and  of  the  location  of  the 
nodal  point  has  been  written  for  a  definite  case,  for  an  east  wind  over  Lake 
Erie,  and  for  an  initial  point  at  the  extreme  eastern  part  of  Lake  Erie,  where 
the  water  will  be  lowest  under  the  influence  of  an  east  wind.  The  demon- 
stration and  the  corresponding  methods  of  computation  are  of  general 
application,  with  certain  obvious  minor  changes  in  statement,  for  any  lake, 
for  any  initial  point  on  that  lake,  and  for  any  wind  direction. 

The  computations  have  been  made  for  eight  wind  directions  for  Lake 
Erie  and  for  Lake  Michigan-Huron. 

As  already  noted  (page  46),  in  making  the  computations  for  an  east 
wind  over  Lake  Erie  it  was  found  that  a  special  auxiliary  computation  of 
the  general  character  shown  in  table  No.  8  must  be  made  for  Sandusky  Bay. 
In  the  corresponding  auxiliary  computation  for  Sandusky  Bay  of  the  charac- 
ter illustrated  by  table  No.  9  the  computation  was  started  with  the  value 

of  S —  found  at  the  entrance  of  the  bay,  at  the  first  strip  in  the  bay 

which  is  found  to  be  cut  off  from  the  corresponding  strip  in  the  main  lake  by 
intervening  land. 

Many  such  auxiliary  computations  were  found  to  be  necessary  for  bays 
cut  off  from  the  main  lake  by  intervening  land,  so  far  as  certain  strips  of 
the  character  used  in  the  computation  are  concerned. 

In  what  precedes,  the  expression  "nodal  point"  has  frequently  been  used, 
having  in  mind  a  profile  view  of  the  water  surface  at  right  angles  to  the 


BAROMETRIC   PRESSURES   ON   THE    GREAT   LAKES  51 

assumed  wind  direction.  Such  a  view  is  indicated  in  figure  1,  plate  4.  A 
nodal  line  passes  through  the  nodal  point  of  the  profile  and  extends  across 
the  lake.  Under  the  action  of  a  wind,  the  nodal  line  on  the  surface  of  the 
water  remains  unchanged  in  elevation,  all  parts  of  the  lake  surface  to  leeward 
of  the  nodal  line  are  raised,  and  all  parts  to  the  windward  are  lowered. 


POSITIONS  OF  VARIOUS  NODAL  LINES. 

The  positions  of  the  various  nodal  lines  for  various  wind  directions  and 
for  Lakes  Erie  and  Michigan-Huron  are  shown  on  plates  2,  5,  and  6.  Each 
nodal  line  appertains  to  a  wind  direction  at  right  angles  to  that  line. 

It  will  be  noted  that  in  various  cases  there  is  a  short  portion  of  a  nodal 
line  extending  across  a  bay,  which  portion  is  parallel  to  but  not  in  line  with 
the  portion  of  the  nodal  line  which  is  in  the  main  lake.  In  each  of  these 

cases  the  value  of  S —  which  identifies  the  nodal  line  was  found  by  exam- 
inations of  the  computation  to  exist  in  the  bay.  These  are  true  nodal  lines 
fixed  at  certain  locations  in  certain  bays,  due  to  the  fact  that  the  slopes  of 
the  water  surface  in  those  bays  are  as  shown  in  formula  (56),  and  the  eleva- 
tion of  the  water  surface  in  the  mouth  of  the  bay,  where  it  is  first  cut  off 
from  the  lake  by  intervening  land,  is  as  indicated  by  formula  (59). 

On  Lake  Erie  (see  plate  2)  the  nodal  line  for  east  and  west  winds,  a  line 
which  therefore  runs  north  and  south,  is  much  nearer  to  the  western  end  of 
the  lake  than  to  its  middle.  It  is  in  the  longitude  of  Pelee  Island.  Note 
that  when  the  wind  becomes  northwest  or  southeast  the  main  portion  of  the 
nodal  line  still  remains  in  this  locality,  running,  of  course,  in  the  southwest- 
northeast  direction,  but  that  it  has  two  detached  bay  portions,  one  in  the 
bight  northeast  of  Point  Pelee  and  the  other  in  the  bight  northeast  of 
Point  aux  Pins.  For  north  winds  and  south  winds  the  main  portion  of  the 
nodal  line  is  in  a  central  position  across  Lake  Erie,  but  a  bay  portion  extends 
westward  from  near  the  end  of  Point  Pelee.  For  northeast  winds  and 
southwest  winds  the  nodal  line  is  again  far  from  the  central  portion  of  the 
lake — in  the  southwest  part.  The  crowding  of  three  of  the  four  nodal  lines 
toward  the  western  end  of  Lake  Erie  is  due  to  the  relative  shallowness  of 
that  portion  of  the  lake  which  is  west  of  Pelee  Island.  There  the  depth  is 
less  than  6  fathoms,  as  a  rule,  whereas  in  much  of  the  main  portion  of  the 
lake  the  depth  is  from  10  to  14  fathoms. 

On  Lake  Michigan-Huron  (see  plates  5  and  6)  the  main  portion  of  the 
nodal  line  for  southwest  winds  and  northeast  winds  lies  in  Lake  Michigan, 
not  far  to  the  westward  of  the  Strait  of  Mackinac,  and  has  two  detached  bay 
portions,  one  across  Saginaw  Bay  and  one  near  Port  Huron.  The  main 
portion  of  the  nodal  line  for  west  winds  or  east  winds  is  in  the  same  locality 
in  Lake  Michigan,  not  far  to  the  westward  of  the  Strait  of  Mackinac,  and 
has  a  single  detached  bay  portion  across  Saginaw  Bay.  The  main  portion 
of  the  nodal  line  for  northwest  winds  and  southeast  winds  is  in  Lake  Huron, 


52  EFFECTS   OF   WINDS   AND    OF 

far  to  the  southeastward  of  the  Strait  of  Mackinac.  It  has  no  detached 
bay  portions  worthy  of  note.  For  north  winds  and  south  winds  the  nodal 
line  consists  of  three  short  detached  portions,  one  near  the  extreme  southern 
end  of  Lake  Michigan,  one  across  Saginaw  Bay,  and  one  in  the  extreme 
southern  end  of  Lake  Huron,  near  Port  Huron.  Note  the  range  horizontally 
through  which  the  nodal  line  shifts  when  the  wind  changes  from  northwest 
to  north  and  again  when  it  changes  from  north  to  northeast. 

There  are  some  other  very  short  detached  bay  portions  of  the  nodal  lines 
on  Lake  Michigan-Huron,  but  they  are  of  local  importance  only.  They 
can  not  be  shown  on  the  scale  of  plates  5  and  6. 

To  apply  formula  (59)  to  a  particular  gage,  the  Buffalo  gage  for  example, 
let  it  be  rewritten  thus: 

(65) 


in  which  the  subscripts  6  stand  for  the  Buffalo  gage,  and  2b  is  the  2—  of 

formula  (59)  from  the  nodal  line  to  the  Buffalo  gage.  The  value  of  2s  may 
be  obtained  easily  from  such  a  computation  as  that  illustrated  in  table  No. 

9  by  merely  taking  the  difference  between  2  —  at  the  nodal  line,  of  which 
the  value  is  determined  from  this  computation  as  already  indicated,  and  the 
value  of  2—  at  the  Buffalo  gage  which  is  shown  directly  in  the  computa- 

tion. 

Formula  (65)  is  of  the  proper  form  for  application  to  any  gage  on  any 
lake  by  merely  changing  the  subscript  and  making  the  corresponding  changes 
in  interpretation. 

The  values  of  the  2  of  such  an  equation  as  (65)  will  in  general  be  different 
for  each  gage  and  for  each  direction  of  wind  over  the  lake. 

VALUES  OF  2*. 

The  various  values  of  2*  have  been  computed,  in  the  manner  indicated  on 
pages  44-52,  for  the  five  gages  at  Buffalo,  Cleveland,  Milwaukee,  Harbor 
Beach,  and  Mackinaw.  The  values  are  shown  in  table  No.  10,  which  follows, 
in  the  units  indicated  in  connection  with  tables  Nos.  8  and  9. 

There  are  certain  features  in  table  No.  10  which  it  is  desirable  to  note  in 
the  attempt  to  secure  definite  and  correct  ideas  as  to  the  wind  effects  on  the 
two  lakes  under  consideration.  Attention  is  called  to  some  of  these  in  the 
following  paragraphs.  Consult  plates  2,  5,  and  6  in  connection  with  these 
paragraphs. 

The  values  of  2*  are  in  general  much  larger  for  the  two  Lake  Erie  gages 
than  for  the  three  Lake  Michigan-Huron  gages.  The  smallest  value  of  2X, 
at  either  Buffalo  or  Cleveland,  shown  in  the  table  is  1.72  for  north  and  south 
winds  at  Cleveland.  The  largest  value  of  2X  at  any  of  the  three  Lake 
Michigan-Huron  gages  is  0  .  95  for  west  and  east  winds  at  Milwaukee.  Hence, 


BAROMETRIC   PRESSURES   ON   THE    GREAT   LAKES 


53 


one  must  expect  to  find  the  wind  effects  much  smaller  at  these  Lake  Mich- 
igan-Huron gages  than  at  these  Lake  Erie  gages. 

TABLE  No.  10. — Values  ofzxfor  use  in  formula  (59)  at  the  gages  indicated. 


Direction  of 
wind. 

At 
Buffalo. 

At 
Cleveland. 

At 
Milwaukee. 

At  Harbor 
Beach. 

At 
Mackinaw. 

NE 

—8  32 

—3  89 

+0  16 

—0  86 

0  64 

E  

-5.90 

-2.86 

+   .95 

—     77 

—     41 

SE 

—2  26 

—  1  90 

+     25 

—     05 

+     15 

S  

+3.45 

-1.72 

+   .36 

+     35 

+     44 

sw.    .  . 

+8  32 

+3  89 

—     16 

+     86 

+    64 

w 

+5  90 

+2  86 

—     95 

+     77 

+     41 

NW. 

+2  26 

+  1  90 

—     25 

+     05 

—     15 

N  

-3.45 

+  1.72 

-    .36 

-    .35 

-    .44 

The  average  values  of  Sx  at  the  five  gages  stand  in  the  order  in  which  the 
gages  are  given  in  table  No.  10.  Buffalo  has  the  largest  average  value  of  2*, 
and  hence  the  largest  wind  effects,  among  these  stations,  and  Mackinaw  has 
the  smallest. 

Buffalo  is  at  the  eastern  end  of  Lake  Erie,  and  Cleveland  is  to  the  westward 
of  the  middle  of  the  lake.  Cleveland  is  on  the  south  shore  of  the  lake,  and 
Buffalo  is  near  the  most  northerly  point  of  the  lake.  From  these  facts 
alone  one  would  infer  that  in  general  the  wind  effects  would  be  of  opposite 
signs  at  the  two  gages.  Yet  table  No.  10  shows  the  wind  effects  to  be  the 
same  in  sign  at  the  two  gages  for  six  out  of  the  eight  directions  tabulated.  The 
only  values  of  S*  at  the  two  gages  which  are  of  opposite  signs  are  those  for 
the  directions  south  and  north.  This  apparently  anomalous  agreement  in 
signs  arises  from  the  crowding  of  the  nodal  lines  into  the  western  portion  of 
Lake  Erie,  in  the  region  of  shallow  water,  to  which  attention  was  called  on 
page  51,  in  such  wise  that  for  all  directions  of  wind  except  north  and  south 
Cleveland  and  Buffalo  are  on  the  same  side  of  the  nodal  line.  The  effect  of 
southwest,  west,  and  northwest  winds  is  to  raise  the  elevation  of  the  water 
surface  at  Cleveland  rather  than  to  lower  it.  One  would  naturally  expect 
the  latter  from  consideration  of  the  plan  of  the  lake  alone. 

There  is  a  large  change  in  Sz  at  Cleveland  when  the  wind  changes  from 
south  to  southwest,  viz.,  from  —  1 . 72  to  +3 . 89.  Note  by  comparison  with 
plate  2  that  this  is  due  to  the  sudden  shift  of  the  nodal  line  from  the  leeward 
of  Cleveland  to  the  windward  when  this  change  of  wind  occurs.  The 
corresponding  statements  are  true  for  the  shift  of  wind  from  north  to  north- 
east. 

Milwaukee  is  much  nearer  the  southern  end  of  Lake  Michigan  than  the 
northern  end.  From  this  fact  alone  one  would  naturally  infer  that  a  south 
wind  would  lower  the  elevation  of  the  water  surface  at  Milwaukee.  But 
table  No.  10  shows  that  Sx  for  Milwaukee  for  a  south  wind  is  +0.36  and 
that  therefore  the  water  surface  is  raised  at  Milwaukee  by  a  south  wind. 


54  EFFECTS   OF   WINDS   AND    OF 

Note  that  this  is  due  to  the  fact  that  the  nodal  line  for  a  south  wind  is  near 
the  extreme  southern  end  of  Lake  Michigan,  as  shown  on  plate  5. 

The  same  apparently  anomalous  condition  is  also  found  at  Harbor  Beach 
(see  plate  6) .  For  a  south  wind  the  nodal  line  is  near  to  Port  Huron,  at  the 
extreme  southern  end  of  Lake  Huron.  S*  for  a  south  wind  is  +0.35  at 
Harbor  Beach,  and  a  south  wind  raises  the  water  surface  at  Harbor  Beach. 

THE  WIND  EXPONENT. 

The  greater  the  velocity  of  the  wind  blowing  over  a  lake  the  more  rapid 
will  be  the  drift  of  the  surface  water  to  leeward.  The  larger  the  surface 
drift  to  leeward  the  greater  will  be  the  return  current  to  windward  ultimately 
produced  by  gravity  after  the  steady  regime  has  been  established.  The 
greater  the  return  current  the  steeper  will  be  the  surface  slopes  of  the  water 
and  the  greater  the  disturbance  of  elevation  of  the  surface  at  any  given  point 
for  a  wind  from  a  given  direction. 

As  a  first  approximation,  it  might  be  assumed  that  the  rate  at  which  the 
wind  delivers  water  to  leeward  in  the  surface  drift  is  proportional  to  the 
velocity  of  the  wind.  If  so,  the  velocity  of  the  return  gravity  current  to 
windward  would  be  proportional  to  the  wind  velocity.  As  shown  by  the 
Chezy  formula,  (52),  page  40,  the  surface  slope  for  a  steady  current  produced 
by  gravity  is  proportional  to  the  square  of  the  velocity  of  the  current.  The 
wind  effects,  as  disturbances  of  elevation  of  the  water  surface  at  a  gage,  are 
proportional  to  the  slopes.  Hence,  on  the  first  approximation  suggested, 
the  wind  effects  would  be  proportional  to  the  squares  of  the  wind  velocities, 
and  the  exponent  of  h  in  the  wind-effect  formula  (51),  page  39,  would  be  2.0. 

But  the  validity  of  the  approximate  assumption  suggested  is  decidedly 
uncertain.  So  far  as  the  writer  knows,  there  is  no  proof,  theoretical  or 
observed,  that  the  drift  of  the  water  to  leeward  at  the  very  surface  is 
proportional  to  the  wind  velocity,  though  that  seems  to  be  a  plausible 
assumption. 

The  depth  to  which  the  surface  drift  extends  is  probably  a  function  of  the 
wind  velocity.  With  higher  wind  velocities,  the  drift  at  the  very  surface 
will  certainly  be  more  rapid,  and  the  depth  to  which  the  drift  extends  will 
probably  be  greater  than  during  light  winds.  If  this  is  the  case,  then  the 
total  drift  to  leeward  in  a  given  wind,  expressed  in  volume  per  unit  of  time, 
the  Q  of  page  40,  will  be  a  different  function  of  the  wind  velocity  than  is  the 
velocity  of  the  drift  at  the  very  surface. 

The  character  of  the  water  surface  on  which  the  wind  acts  to  drive  the 
water  to  leeward  varies  greatly  for  different  wind  velocities.  It  varies 
greatly  in  roughness,  and  the  roughnesses  are  themselves  in  motion  at  rates 
which  are  in  various  relations  to  the  wind  velocities.  During  very  light 
winds  the  water  surface  is  relatively  very  smooth,  broken  only  in  general  by 
wind  ripples.  With  moderate  winds,  say  10  to  15  miles  per  hour,  there  is  a 
decided  roughness  in  the  form  of  wind  waves,  which  are  traveling  to  leeward 
at  a  rate  not  differing  greatly  from  the  velocity  of  the  wind.  The  wind  has  a 


BAROMETRIC    PRESSURES   ON   THE    GREAT   LAKES  55 

much  lighter  grip  upon  or  impact  upon  these  roughnesses  than  it  would  have 
if  the  roughnesses  were  stationary.  The  impact  of  the  wind  on  the  rear 
surface  of  a  moving  wave  is  certainly  rather  light  when  the  wave  is  moving 
nearly  as  fast  as  the  wind.  During  very  high  winds,  say  in  excess  of  40 
miles  per  hour,  the  wind  waves  are  high  and  expose  a  very  rough  surface  to 
the  action  of  the  wind,  this  roughness  is  traveling  to  leeward  at  a  much 
slower  rate  than  the  wind  moves,  and  the  drift  of  water  to  leeward  is  now 
augmented  by  the  large  throw  to  leeward  of  the  upper  part  of  each  wave  as 
it  breaks. 

From  such  considerations  as  are  indicated  in  the  preceding  few  paragraphs 
it  appears  that  the  wind  exponent  in  such  a  formula  as  (51)  is  probably  2 
(or  more),  but  that  the  theory  is  so  uncertain  that  it  is  best  to  derive  the 
exponent  from  observations  rather  than  from  theory. 

Accordingly,  in  early  stages  of  the  investigation  of  wind  effects,  a  number 
of  least-square  solutions  were  made  in  pairs,  in  which  the  two  solutions  of  a 
pair  differed  only  or  mainly  in  the  exponent  assigned  to  h,  the  wind  velocity. 
These  solutions  indicated  that  the  more  nearly  the  exponent  was  made  to 
approach  to  2.4  from  either  side  the  closer  was  the  agreement  obtained  be- 
tween the  computed  results  and  the  observed  facts.  In  other  words,  more 
accurately  stating  the  matter,  the  sum  of  the  squares  of  the  residuals  ap- 
peared to  be  a  minimum  when  the  exponent  2.4  was  used.  A  residual  in 
this  statement  is  a  discrepancy  between  the  computed  change  of  elevation 
of  the  water  surface  at  a  gage  during  a  given  interval,  on  the  one  hand,  and 
the  observed  change  of  elevation  recorded  by  the  gage  during  that  interval, 
on  the  other  hand. 

Hence,  the  exponent  2.4,  derived  thus  from  observations,  was  adopted  for 
the  final  formula  for  wind-effect  investigations,  as  shown  in  (51),  page  39. 

Some  further  information  is  given  at  an  appropriate  place  later  in  this 
publication,  in  the  discussion  of  errors  of  computed  wind  effect,  as  to  the 
particular  least-square  solutions  which  served  to  establish  2.4  as  the  most 
probable  value  of  the  exponent  and  as  to  the  estimated  accuracy  of  that 
exponent. 

At  this  point  attention  is  called  especially  to  the  fact  that  the  value  2.4 
is  derived  from  observation  rather  than  theory.  It  is  the  investigator's  be- 
lief, based  on  much  study  of  the  details  of  the  investigation  and  some  theo- 
retical considerations,  that  an  error  of  moderate  amount  in  the  adopted 
exponent  has  but  slight  effect  on  the  final  outcome  of  the  investigation 
expressed  in  terms  of  daily  corrections  for  wind  effect,  provided  the  expo- 
nent, once  adopted,  is  used  consistently  throughout  the  remainder  of  the 
investigation. 

EXAMPLE  OF  OBSERVATION  EQUATIONS  FOR  WIND  EFFECTS. 

The  least-square  solutions  for  determining  the  wind  effects  are  based 
upon  hourly  observations  of  the  water  surface  and  upon  formula  (51), 
shown  on  page  39. 


56  EFFECTS   OF   WINDS   AND    OF 

The  form  of  each  observation  equation  is  as  follows: 


h  is  the  wind  velocity,  in  miles  per  hour,  at  the  station  to  which  the  equation 
refers,  ending  at  the  hour  specified  by  the  subscript. 

The  subscript  c  refers  to  the  current  hour  —  the  hour  by  which  the  equa- 
tion is  designated.  The  subscript  p  refers  to  the  preceding  hour.  The 
subscript  c+1  refers  to  the  hour  following  the  current  hour. 

Cp  and  Ca  are  unknown  constants  to  be  derived  from  the  observations  by 
means  of  the  least-square  solution. 

L  is  the  elevation  of  the  water  surface  at  the  gage  at  the  current  hour 
minus  the  elevation  of  the  water  surface  at  the  gage  at  the  preceding  hour, 
the  elevations  being  first  corrected  for  any  known  effects  for  which  it  is 
feasible  to  apply  reliable  corrections.  The  only  such  correction  applied  in 
this  investigation  was  the  correction  for  hourly  barometric  effects,  computed 
as  indicated  on  pages  32-36. 

The  symbol  Si  stands  for  the  appropriate  value  from  table  No.  10  for  the 
gage  station  under  consideration  —  Buffalo  gage,  Cleveland  gage,  etc.  —  and 
for  the  wind  direction  at  that  station  for  the  hour  specified  by  the  subscript 

(/i2'4\ 
-  ]  into  which  this  particular  S*  is  multiplied. 

Compare  (66)  with  (51)  on  page  39.  It  appears  that  if  Cp,  an  unknown 
constant  to  be  determined,  is  considered  to  be  the  Cx  of  equation  (51),  then 

(  -  )  (2*)CP  in  (66)  is  the  effect  of  the  wind  at  the  preceding  hour  in  elevat- 
\100/  p 

(/i2'4\ 
-  ((S^Cp,  on  the 
100/c 

same  supposition,  is  the  effect  of  the  wind  at  the  current  hour  in  elevating 
the  surface  of  the  water  at  the  gage.  The  difference  shown  as  the  first  of 
three  terms  in  the  first  member  of  the  observation  equation  (66),  namely, 


is  the  computed  fall  in  the  water  surface  from  the  preceding  to  the  current 
hour  on  the  supposition  that  Cp  is  the  Cx  of  equation  (51) . 

One  modification  should  be  made  of  the  statement  in  the  preceding 
paragraph,  in  which  it  is  implicitly  assumed  that  there  is  no  lag  in  the  re- 
sponse of  the  water  to  a  change  of  wind.  It  should  be  noted  that  h,  as 
defined  just  below  equation  (66),  is  the  wind  velocity  at  the  hour  ending  at 
the  time  specified.  If,  for  example,  the  hour  specified  is  10  a.m.,  the  velocity 
h  is  the  velocity  for  the  hour  from  9  a.m.  to  10  a.m.  This  velocity  is  in 
fact  ordinarily  obtained  by  counting  up  the  number  of  miles  of  wind,  as 
shown  on  an  automatic  record  from  an  anemometer,  that  passed  the  record- 


BAROMETRIC   PRESSURES   ON   THE    GREAT   LAKES  57 

ing  station  between  9  a.m.  and  10  a.m.  In  such  a  case  the  velocity  applies 
more  strictly  to  9.30  a.m.  than  to  10  a.m.,  or,  in  other  words,  it  is  the  velocity 
of  one-half  hour  before  the  time  specified.  So  the  statement  of  the  preceding 
paragraph  would  apply  strictly  if  the  wind  effect  lagged  one-half  hour 
behind  each  change  in  the  wind. 

Compare  the  second  of  the  three  terms  in  the  first  member  of  observation 
equation  (66),  namely, 

— )  (Ex)  —  I  — }      (£*)    C, 


with  equation  (51).  It  appears  from  the  comparison  that  in  the  same 
manner  that  the  first  term  expresses  the  computed  fall  in  the  water  surface 
if  the  lag  is  one-half  an  hour,  so  the  second  term  would  properly  express  it  if 
instead  of  lag  there  is  an  anticipation  of  one-half  an  hour.  An  anticipation 
means  in  this  case  merely  that  the  water  surface  at  the  gage  station  changes 
before  the  change  in  wind  occurs  at  the  Weather  Bureau  station  at  which 
the  wind  is  recorded,  which  may  be  a  mile  or  even  several  miles  away,  and 
not  that  the  effect  on  the  water  preceded  the  cause  which  produced  it.  It  is 
desirable,  also,  if  one  tends  to  be  skeptical  of  an  anticipation  in  the  sense 
indicated  in  the  formula,  to  consider  that  an  effect  at  the  gage  may  precede 
the  arrival  of  the  wind  change  at  the  gage,  since  the  wave  of  water  piled  up 
by  an  approaching  wind  may  outrun  the  progressive  change  in  the  wind. 

If,  then,  the  least-square  solution  shows  derived  values  of  Cp  and  Ca  which 
are  equal,  the  meaning  is  that  the  wind  effect  at  the  gage  (a  change  of 
elevation  of  water  surface)  is,  upon  an  average,  simultaneous  with  the  change 
in  the  wind  at  the  Weather  Bureau  station,  if  Cp  is  finite  and  Ca  zero  the  lag 
is  0.5  hour,  and  if  Cp  is  zero  and  Ca  finite  there  is  an  anticipation  of  0.5 
hour.  For  intermediate  cases  the  lag  or  anticipation  has  intermediate 
values. 

Studies  and  various  least-square  solutions  made  in  this  investigation  be- 
fore the  final  form  of  the  observation  equations  as  shown  in  (66)  was  adopted 
showed  that  the  discoverable  lag,  if  any,  in  the  wind  effects  is  probably  very 
small,  a  few  minutes  only.  Hence,  (66),  based  on  the  supposition  that  the 
lag  is  very  small,  is  deemed  to  be  the  best  form  for  the  observation 
equations. 

If  the  computed  fall  of  the  water  surface,  represented  by  the  first  two 
terms  of  the  first  member  of  equation  (66),  were  exactly  equal  to  and  opposite 
in  sign  to  the  observed  rise,  L,  then  the  whole  first  member  of  (66)  would  be 
zero,  and  the  computed  residual,  v,  in  the  second  member  would  be  zero. 
This  would  be  the  case  if  both  the  theory  and  all  the  observations  were 
absolutely  perfect.  In  the  actual  case,  each  v,  a  discrepancy  between 
computation  and  observation,  is  a  residual  for  a  particular  hour  between 
theory  and  computation  on  the  one  hand  and  observation  on  the  other.  A 
large  group  of  such  residuals  from  many  observation  equations  furnishes  a 
measure  of  the  accuracy  of  the  theory  and  the  computation  based  upon  it. 


58  EFFECTS   OF   WINDS   AND    OF 

The  following  set  of  observation  equations  for  August  5,  1910,  at  Buffalo, 
serves  as  a  typical  example.  They  are  a  part  of  solution  W25,  which  included 
in  all  470  such  equations,  covering  500  hours  out  of  22  selected  days. 

Wind  Observation  Equations,  Buffalo,  August  5,  1910,  Solution  W25. 

la.m -  33CP-     6C0-29  =  t>i  1  p.m -150CP-100Ca+20  =  ru 

2          -     6CP-  24Ca+14  =  t>2  2          .  .-lOOCp        OC0+  8  =  t>u 

3-4      -  47CP-  llCa+31=t>,  3          OCP-  33C0-  5=wu 

5-6      -  12CP-  24Ca+32  =  r4  4          -  33CP-  83Ca+  2  =rw 

7          OCP        OC0-  3  =  r8  5          -  83Cp+  83Ca+17  =  t>u 

8          OCP+  24C0-16  =  t>6  6          +  83CP-  42C0+19  =  rls 

9          +  24CP-  48C0-14  =  W7  7          -  42CP+  75Ca-21=t>j7 

10          -  48CP-  70C0+23  =  rg  8          +  75CP+374C0+  2  =  t;18 

11  a.m -  70Cp-160Cffl-21  =  t>9  9          +374CP  +  154Ca-30  =  n, 

N -160CP-150Ca +22  =  t*o          10          +  154CP+  59Ca-27=v» 

11  p.m +  59CP-  59Ca+26=t>n 

M -  59CP-  12C0+ll=tfe 

The  unit  used  in  expressing  L,  the  absolute  term,  is  0.01  foot. 

The  basis  on  which  such  combinations  as  are  indicated  for  3  and  4  a.m. 
and  5  and  6  a.m.  were  decided  upon  will  be  indicated  later  in  connection 
with  the  discussion  of  the  accuracy  of  the  computed  wind  effects.  The  basis 
for  certain  rejections  which  were  made  will  also  be  indicated  later  in  the 
same  place. 

Table  No.  11,  which  follows  on  page  59,  shows  how  the  coefficients  of 
Cp  and  Ca  were  computed. 

The  wind  velocities  and  wind  directions  as  shown  in  the  second  and 
third  columns  of  table  No.  11  were  observed  at  Buffalo  by  the  Weather 
Bureau. 

The  values  in  the  second  column,  wind  velocities  in  miles  per  hour,  are  the 
values  of  h  from  which  the  fourth  column  of  table  No.  11  was  computed. 

The  values  of  S&  shown  in  the  fifth  column  were  taken  from  table  No.  10 
for  the  observed  wind  directions  at  Buffalo  as  shown  in  the  third  column. 

Each  value  in  the  sixth  column  is  the  product  of  the  values  shown  on  the 
same  line  in  the  fourth  and  fifth  columns. 

Each  value  of  the  coefficient  of  Cp  as  shown  in  the  seventh  column  is  the 
difference,  in  the  sense  (preceding  —  current) ,  of  the  values  shown  on  two 
lines  in  the  sixth  column.  Similarly,  the  coefficient  of  Ca  as  shown  in  the 
last  column  of  the  table  is  the  difference  of  two  values  in  the  sixth  column. 

A  comparison  of  the  seventh  and  eighth  columns  of  table  No.  11  with  the 
coefficients  of  Cp  and  C0  in  the  example  of  observation  equations  will  make 
the  relation  clear.  Note  that  in  the  combined  equation  for  the  two  hours 
3  and  4  a.m.  the  coefficient  of  CP,  —47,  is  the  algebraic  sum  of  the  two  values 
for  the  coefficient  of  Cp  shown  in  table  No.  11  for  the  hours  3  and  4  a.m., 
namely,  —24  and  —23,  respectively.  In  each  such  case  the  coefficient  in  a 
combined  equation  is  the  algebraic  sum  of  the  corresponding  coefficients 
of  the  separate  equations  which  were  combined. 


BAROMETRIC   PRESSURES   ON   THE    GREAT   LAKES  59 

TABLE  No.  11.— Buffalo,  August  5,  1910. 


Wind 
velocity. 

Wind 
direction. 

If.:* 
100 

S& 

D-> 

Required  coefficients. 

CP 

Ca 

W. 

M  

14 

NW 

6 

+2.26 

+  14 

feet. 

1  a.m  

16 

W 

8 

+5.90 

+  47 

-  33 

-'  6 

+"04 

2          

17 

w 

9 

+5.90 

+  53 

-     6 

-  24 

+  .05 

3          

20 

W 

13 

+5.90 

+  77 

-  24 

-  23 

+  .07 

4          

22 

w 

17 

+5.90 

+100 

-  23 

+  12 

+  .09 

5          

21 

w 

15 

+5.90 

+  88 

+  12 

-  24 

+  .08 

6          

23 

w 

19 

+5.90 

+112 

-  24 

0 

+  .10 

7          

23 

w 

19 

+5.90 

+112 

0 

0 

+  .10 

8 

23 

w 

19 

+5.90 

+112 

0 

+  24 

+  .10 

9         

21 

w 

15 

+5.90 

+  88 

+  24 

-  48 

+  .08 

10 

25 

w 

23 

+5.90 

+136 

-  48 

-  70 

+  .12 

11  a.m  

30 

w 

35 

+5.90 

+206 

-  70 

-160 

+  .18 

N  

33 

sw 

44 

+8.32 

+366 

-160 

-150 

+  .32 

1  p.m  

38 

sw 

62 

+8.32 

+516 

-150 

-100 

+  .45 

2          

41 

sw 

74 

+8.32 

+616 

-100 

0 

+  .54 

3          

41 

sw 

74 

+8.32 

+616 

0 

-  33 

+  .54 

4          

42 

sw 

78 

+8.32 

+649 

-  33 

-  83 

+  .57 

5          

44 

sw 

88 

+8.32 

+732 

-  83 

+  83 

+  .64 

6          

42 

sw 

78 

+8.32 

+649 

+  83 

-  42 

+  .57 

7          

43 

sw 

83 

+8.32 

+691 

-  42 

+  75 

+  .61 

8         

41 

sw 

74 

+8.32 

+616 

+  75 

+374 

+  .54 

9          

32 

w 

41 

+5.90 

+242 

+374 

+  154 

+  .21 

10         

21 

w 

15 

+5.90 

+  88 

+154 

+  59 

+  .08 

11  p.m  

20 

NW 

13 

+2.26 

+  29 

+  59 

-  59 

+  .03 

M 

21 

w 

15 

+5.90 

+  88 

-  59 

-  12 

+  .08 

Jj2-4 

Note  that  in  table  No.  11  was  88  at  5  p.m.,  when  the  wind  velocity 

-LU(J 

was  44  miles  per  hour,  and  only  6  at  the  preceding  midnight,  when  the  wind 

velocity  was  14  miles  per  hour — that  is,  was  increased  nearly  fifteen- 

1UU 

fold  between  midnight  and  5  p.m.  The  increase  in  the  theoretical  wind 
effect  would  therefore  have  been  nearly  fifteenfold  if  there  had  been  no 
change  in  wind  direction.  With  the  actual  change  in  wind  direction  which 
occurred,  from  NW  to  SW,  the  theoretical  wind  effect  increased  more  than 
fifty-two  times,  as  the  change  in  the  sixth  column  of  table  No.  11  is  from  14 
to  732.  Attention  is  invited  to  this  case,  and  in  general  to  an  inspection  of 
table  No.  11,  to  secure  an  appreciation  of  the  complicated  law  controlling 
the  wind  effects.  Note  the  relatively  large  increases  in  wind  effects  produced 
by  moderate  increases  in  high  winds,  as,  for  example,  the  increase  from  42  to 


60 


EFFECTS   OF   WINDS   AND   OF 


44  miles  per  hour  between  4  and  5  p.m.  Note,  also,  the  large  changes 
produced  by  a  change  of  45°  in  the  wind  direction. 

The  last  column  of  table  No.  11,  headed  W,  is  explained  later.  It  is  not 
intended  for  consideration  at  this  point. 

The  observed  elevations  of  the  water  surface  at  the  Buffalo  gage  at  each 
hour,  as  read  from  the  automatic  gage  record  of  the  Lake  Survey,  the 
hourly  barometric  effects  as  computed  according  to  pages  32-36,  and  the 
corrected  elevation  of  the  water  surface  at  each  hour  as  it  would  have  been 
if  no  barometric  effect  had  occurred  are  shown  below. 

TABLE  No.  HA.— Buffalo,  August  5,  1910. 


Observed  elevation 
of  water  surface. 

Computed 
barometric  effect. 

Corrected  elevation 
of  water  surface. 

1  a.m  

feet. 
572  12 

feet. 
+  26 

/•* 

571  86 

2 

572  26 

+  26 

572  00 

3         

572  74 

+  27 

572  47 

4 

572  60 

+  29 

572  31 

5          

572  49 

+  29 

572  20 

6          

572  92 

+  29 

572  63 

7 

572  90 

+  30 

572  60 

8          

572  76 

+  32 

572  44 

9 

572  62 

+  32 

572  30 

10          

572  86 

+  33 

572  53 

11  am 

572  67 

+  35 

572  32 

N.. 

572  90 

+  36 

572  54 

1  p.m  

573  12 

+  38 

572  74 

2 

573  18 

+  36 

572  82 

3          

573  12 

+  35 

572  77 

4          
5          
6          

7         
8          

9 

573.13 
573.30 
573.48 
573.28 
573.28 

572  96 

+  .34 
+  .34 
+  .33 
+  .34 
+  .32 

+  30 

572.79 
572.96 
573.15 
572.94 
572.96 

572  66 

10          
11  p.m  

572.68 
572  39* 

+  .29 
+  .26 

572.39 
572  13 

M 

572  49 

+  25 

572  24 

The  observed  elevations  are  referred  to  mean  sea-level. 

The  values  of  L,  see  equation  (66) ,  page  56,  as  recorded  in  the  observation 
equations  shown  on  page  58,  in  units  of  0 . 01  foot,  were  obtained  by  sub- 
traction of  adjacent  values  of  the  corrected  elevations  shown  above.  The 
absolute  term  L,  in  each  of  the  two  observation  equations,  for  the  hours 
3  and  4  a.m.  and  for  the  hours  5  and  6  a.m.,  is  the  corrected  rise  for  two  hours 
instead  of  one. 


BAROMETRIC  PRESSURES  ON  THE  GREAT  LAKES      61 

The  observation  equations  were  arranged  in  groups,  one  group  for  each 
day.  The  groups  were  studied  separately  as  well  as  in  combination  with 
each  other  as  a  single  set  of  equations  for  one  whole  solution. 

EXAMPLES  OF  NORMAL  EQUATIONS  FOR  WIND  EFFECTS^ 

In  each  least-square  solution  for  wind  effects  a  set  of  normal  equations 
was  first  formed  in  the  usual  way  from  the  observation  equations  for  each 
day.  Then  these  sets  of  normal  equations,  one  set  for  each  day,  were  com- 
bined by  addition  to  form  the  final  set  of  normal  equations  for  the  whole 
solution. 

The  normal  equations  for  August  5,  1910,  in  solution  W25,  formed  from 
the  observation  equations  for  that  date,  as  shown  on  page  58,  are  as  follows: 


+252168CP+ 134728Ca  -  21619 
+134728Cp+260143Ca-12627 


=  0  \ 
=  0/ 


The  final  normal  equations  for  solution  W25  for  22  days,  formed  by 
combining  22  such  sets  of  normal  equations  as  are  shown  in  (67),  are  as 
follows  : 


+2914495Cp+  5881  19Ca-  205977  =  0  1 
+  588119Cp+3343518Ca-178350  =  0  J 

This  final  set  of  normal  equations  for  solution  W25,  (68),  depends  upon 
470  observation  equations,  covering  500  hours,  which  formed  parts  of  22 
days. 

The  solution  of  the  final  normal  equations  for  solution  W25,  (68),  gave 
the  following  values  for  the  unknowns: 

Cp  =  +0.0622±.0064  C0=+0.0425±.0060 

The  probable  errors  shown  were  computed  rigorously  from  these  normal 
equations  and  from  the  residuals,  v,  of  the  470  observation  equations  of 
solution  W25. 

THE  FOUR  FINAL  WIND   SOLUTIONS. 

In  this  investigation,  made  by  methods  outlined  on  pages  6-8,  several 
separate  least-square  solutions  were  usually  made  on  the  basis  of  each 
group  of  observational  data.  The  form  of  each  successive  solution  from 
one  group  of  data  was  based  upon  all  the  information  available  up  to  the 
time  that  said  form  was  adopted,  including  the  information  from  earlier 
solutions  based  upon  the  same  data. 

The  four  final  solutions  for  wind  effects  were  those  designated  as  solutions 
W25,  W26,  W20,  and  W29.  Each  of  these  solutions  was  the  culminating 
one  of  a  series  of  solutions  based  on  the  same  group  of  observational  data. 
The  principal  facts  for  these  four  solutions  are  shown  in  tabular  form 
following: 


62  EFFECTS   OF   WINDS   AND   OF 

TABLE  No.  12. — Principal  facts  for  four  final  wind  solutions. 


Solution 
W25. 

Solution 
W26. 

Solution 
W20. 

Solution 
W29. 

Gage  at  which  observations  were 
made  
No.  of  days  of  observation  used 
in  the  solution  
Total  No.  of  hours  of  observation 
available  

Buffalo. 
22 
519 

Buffalo. 
26 
622 

Cleveland. 
22 
519 

Cleveland. 
30 
717 

No.  of  hours  covered  by  the  ob- 
servation    equations     finally 
used                         

500 

614 

508 

669 

No.  of  observation  equations.    . 
Computed  value  of  Cp  
Computed  probable  error  of  Cp  . 
Computed  value  of  Ca  
Computed  probable  error  of  Ca  .  . 
Probable  error  of  a  single  obser- 
vation 

470 
+  .0622 
±.0064 
+  .0425 
±.0060 

±  107ft. 

583 
+  .0733 
±.0106 
-.0008 
±.0095 

±  085  ft. 

490 
+  .0237 
±.0110 
+  .0358 
±.0114 

±  058  ft. 

630 
+  .0700 
±.0122 
+  .0090 
±.0126 

±  052  ft 

The  probable  errors  of  Cp  and  Ca  as  shown  were  computed  rigorously  from 
the  normal  equations  and  the  residuals  of  the  observation  equations. 

The  probable  error  of  a  single  observation  as  shown  above  is  the  probable 
error,  computed  rigorously  from  the  normal  equations  and  the  residuals,  of 
the  observed  change  in  elevation  of  the  water  surface  at  the  gage  in  any  one 
hour. 

As  indicated  on  page  57,  in  connection  with  an  exposition  of  the  meaning 
of  the  quantities  CP  and  C0,  if  Cp  and  Ca  are  equal  there  is  no  lag  in  the  change 
of  elevation  of  the  water  surface  at  the  gage  behind  the  change  in  the  wind 
at  the  Weather  Bureau  station.  If  Cp  is  finite  and  Ca  zero,  the  lag  is  30 
minutes,  and  if  CP  is  zero  and  C0  finite  there  is  an  anticipation  of  30  minutes. 
An  examination  of  the  values  of  Cp  and  Ca  in  table  No.  12  shows  that  solution 
W26  indicates  a  lag  of  30  minutes,  Ca  being  practically  zero;  that  solutions 
W25  and  W29  each  indicate  a  lag  of  less  than  30  minutes;  and  that  solution 
W20  indicates  an  anticipation  or  negative  lag  of  much  less  than  30  minutes. 
If  one  takes  into  account  the  degree  of  uncertainty  indicated  by  the  probable 
errors,  as  well  as  the  smallness  of  the  indicated  lag  in  each  of  the  four  cases 
and  the  discrepancy  between  the  four  indications,  the  conclusion  reached 
is  that  the  actual  lag  is  probably  less  than  30  minutes  and  is  too  small  to  be 
determined  with  reliability  from  the  observations  used.  This  conclusion 
that  the  lag  is  less  than  30  minutes,  and  certainly  too  small  to  be  determined 
from  these  observations,  is  confirmed  by  an  examination  of  many  details  of 
the  evidence. 

In  later  portions  of  the  investigation  the  lag  in  the  wind  effects  was  there- 
fore assumed  to  be  zero.  All  corrections  for  wind  effect  were  computed  on 
that  basis.  On  that  basis  the  Cx  of  the  fundamental  formula  for  wind  effects 
is  Cp+Ca. 


BAROMETRIC   PRESSURES   ON   THE    GREAT   LAKES 


63 


Each  solution  gives  one  value  for  Cz  =  Cp+Ca. 

The  probable  error  of  each  value  of  Cx  from  one  solution  is,  with  sufficient 
accuracy  for  the  present  purpose,  the  square  root  of  the  sum  of  the  squares 
of  the  separate  probable  errors  of  Cp  and  Ca  from  that  solution. 

Table  No.  13  shows  the  four  values  of  Cx  and  their  probable  errors,  the 
weight  assigned  to  each  value  of  Cx,  the  weighted  mean  and  its  probable 
error,  and  the  residuals  of  the  four  separate  values  from  this  weighted  mean. 
The  assigned  weights  are  inversely  proportional  to  the  squares  of  the  probable 
errors  of  Cz,  corresponding  to  the  assumption  that  all  errors  in  computed 
values  of  Cx  are  of  the  accidental  character.  Unit  weight  corresponds  to  a 
probable  error  squared  of  0.001. 

TABLE  No.  13. 


Probable 
error  of 

Cx. 

Assigned 
weight. 

C*  =  CP+Co. 

Residual  from 
weighted 
mean. 

Solution  W25,  Buffalo. 

±0  0087 

13  1 

+0  105 

-0  017 

Solution  W26,  Buffalo  

±   .0142 

4.9 

+   .072 

+   .016 

Solution  W20,  Cleveland  
Solution  W29,  Cleveland  

db   .0158 
±   .0175 

4.0 
3.2 

+   -060 
+   .079 

+   .028 
+   .009 

25.2 

+0.088 

Sum  of  weights  25.2.     Weighted  mean  +0.088.     Probable  error  of  weighted  mean=±0.006. 

The  value  +0 . 088±0 . 006  for  Cx  is  adopted  as  the  best  that  can  be  derived 
from  the  present  investigation.  Accordingly,  equation  (51),  expressing 
the  total  effect  of  the  wind  upon  the  elevation  of  the  water  surface  at  any 
gage,  may  now  be  written  thus: 


W-- 


+o-°88Qs* 


(69) 


In  (69),  W  is  expressed  in  units  of  0.01  foot,  h  is  the  wind  velocity  at  the 
gage  as  determined  by  observations  at  the  nearest  Weather  Bureau  station 
which  makes  such  observations.  Sz  is  a  quantity  dependent  upon  the 
location  of  the  gage  with  reference  to  the  lake,  the  shape  of  the  lake,  the 
depth  in  every  part  of  the  lake,  and  the  direction  of  the  wind.  The  meaning 
of  2X  and  the  manner  of  computing  it  are  fully  stated  on  pages  43-54. 


COMPUTATION  OF  WIND  EFFECTS. 

The  wind  effects  at  each  hour  at  any  gage  might  be  computed  by  the  use 
of  formula  (69)  by  making  the  proper  substitutions  for  each  hour  of  h  and 
Sz.  Such  a  substitution  might  be  made  systematically  by  a  tabulation  such 
as  the  first  six  columns  of  table  No.  11  on  page  59,  supplemented  by  the 
last  column  of  that  table,  which  is  headed  W.  Each  value  in  that  last 


64  EFFECTS   OF   WINDS   AND    OF 

column  is  the  W  of  formula  (69),  namely,  the  value  in  the  sixth  column  of 
the  table  multiplied  by  +0.088. 

The  mean  of  the  24  values  of  W  for  August  5,  1910,  at  Buffalo  gage,  as 
shown  in  the  last  column  of  table  No.  11,  is  +0.258  foot,  which  is  therefore 
the  daily  wind  effect  for  that  day.  The  daily  wind  effects  might  be  so 
computed. 

Merely  as  devices  to  save  time  in  the  computation  of  daily  wind  effects, 
it  was  found  to  be  advisable  to  make  two  changes  in  the  method  of  computa- 
tion which  is  indicated  above.  One  change  was  to  construct  a  table  for 
each  gage,  giving  the  values  of  W  in  terms  of  the  two  arguments,  wind 
velocity  and  wind  direction.  Note  that  in  formula  (69)  there  are  only  two 
variables  in  the  second  member,  the  wind  velocity,  h,  and  2X,  which  is 
variable  for  a  given  gage  as  a  function  of  the  wind  direction  only.  The 

W 
other  change  was  to  make  the  actual  values  placed  in  this  table  —  instead  of 

W  itself.  With  the  table  before  one  and  with  the  observed  wind  velocity 
and  direction  for  each  hour  also  before  one,  the  tabular  values,  one  for  each 
hour,  could  be  taken  out  very  rapidly  to  the  nearest  thousandth  of  a  foot. 
The  sum  of  the  24  tabular  values  for  any  day  was  the  daily  wind  effect.  No 
division  by  24  was  necessary,  as  would  otherwise  have  been  the  case,  be- 
cause each  tabular  value  was  itself  the  result  of  such  a  division. 

ACCURACY  OF  COMPUTED  BAROMETRIC  EFFECTS. 

Having  set  forth  the  theory  and  the  methods  of  computation  by  which 
the  formulae  and  constants  for  computing  barometric  effects  and  wind  effects 
have  been  determined,  it  is  now  proposed  to  set  forth  the  main  portions  of  the 
available  evidence  as  to  the  accuracy  and  reliability  with  which  the  baro- 
metric effects  and  wind  effects  may  be  computed  by  the  use  of  these  formulae 
and  constants. 

It  is  the  purpose  to  set  forth  the  evidence,  first,  in  connection  with  the 
computed  barometric  effects;  second,  in  connection  with  the  computed 
wind  effects;  and,  third,  as  to  the  overall  accuracy  attained  when  corrections 
are  applied  to  each  day's  observations  at  each  gage  for  both  barometric  and 
wind  effects. 

Let  the  evidence  be  considered  here  as  to  the  accuracy  of  the  computed 
barometric  effects. 

Table  No.  6,  page  32,  shows  each  of  the  values  of  the  various  barometric 
constants  BWQ,  Bw\,  etc.,  and  its  probable  error  as  computed  rigorously  from 
the  normal  equations  and  the  residuals  of  the  least-square  solution.  These 
probable  errors  are  a  measure  of  the  accuracy  which  is  the  best  that  can  be 
obtained,  provided  the  errors  in  the  derived  constants  are  all  accidental  in 
character.  Assuming  that  the  errors  are  all  accidental,  that  is,  that  there 
are  no  systematic  or  constant  errors  affecting  the  final  results,  it  is  an  even 
chance  that  the  actual  error  in  any  constant  is  greater  than  or  less  than  its 


BAROMETRIC  PRESSURES  ON  THE  GREAT  LAKES      65 

computed  probable  error  as  shown.  For  example,  £n2  for  Buffalo  =  - 11.69, 
and  its  probable  error  is  only  ±0. 43.  This  means  that  the  chances  are  even 
for  and  against  the  proposition  that  the  true  value  of  BnZ  for  Buffalo  lies 
within  0. 43  of  — 11 . 69.  In  other  words,  there  is  one  chance  in  two  that  the 

value  — 11 . 69  is  correct  within  one  twenty-seventh  of  itself  ( — — -  =  27 ). 

If  for  each  of  the  five  stations  this  comparison  be  made  between  the 
largest  constant  for  the  station  and  its  probable  error,  it  will  be  found  that 
in  each  case  the  probable  error  is  as  small  as  one-twentieth  part,  or  5  per  cent 
of  the  value.  The  largest  constant  is  selected  because  it  is  the  one  which 
tends  to  have  most  influence  upon  the  computed  barometric  effect  at  the 
station. 

On  this  basis  the  conclusion  is  that  the  computed  barometric  constants 
are  subject  to  errors  which  stand  one  chance  in  two  of  being  less  than  5 
per  cent  as  large  as  the  largest  one  of  said  constants  at  each  station. 

Note  that  the  computed  probable  errors  at  each  station  are  of  approxi- 
mately the  same  size  for  all  constants,  regardless  of  the  size  of  the  constant 
itself.  The  extreme  uncertainty,  judged  by  the  probable  errors,  occurs 
in  the  constant  Bn3  for  Harbor  Beach,  of  which  the  value  is  +0. 47,  only  1 . 8 
times  its  own  probable  error,  ±0 . 26.  On  the  supposition  that  all  the  errors 
are  accidental  in  character,  there  is  less  than  one  chance  in  four  that  this 
derived  value,  +0.47,  for  Bn3  at  Harbor  Beach  is  entirely  fictitious.  For 
the  few  cases  in  table  No.  6  like  this  one,  in  which  there  is  a  small  chance 
that  the  constant  is  entirely  fictitious,  the  main  reliance  must  be  placed 
on  other  tests  than  that  furnished  by  the  probable  error  when  one  is  at- 
tempting to  determine  the  accuracy  and  reliability. 

If  in  making  any  least-square  solution  a  large  number  of  rejections  is 
made  of  observations  which  show  large  residuals,  a  fictitious  appearance  of  a 
high  degree  of  accuracy  may  thereby  be  given  to  the  remaining  observations, 
which  necessarily  agree  more  closely  with  each  other  than  did  the  original 
observations  before  any  rejections  were  made.  That  danger  has  been 
guarded  against  in  this  investigation  by  the  adoption  of  a  cautious  rejection 
limit  and  by  a  careful  study  of  each  rejected  value  to  determine  from  ex- 
ternal evidence  if  possible  whether  the  value  may  properly  be  rejected.  The 
external  evidence  has  in  a  large  percentage  of  cases  been  clearly  in  favor  of 
the  rejection.  It  is  believed,  therefore,  that  there  is  no  danger  that  any 
fictitious  accuracy  has  been  introduced  by  the  rejections. 

THE  REJECTION  RULE. 

The  rejection  rule  has  been  that  an  observation  shall  be  rejected  if  its 
residual  is  larger  than  five  times  the  probable  error  of  the  observation,  and 
that  no  other  observations  shall  be  rejected. 

If  all  the  errors  were  strictly  accidental  in  character,  this  rule  would 
reject  less  than  one  observation  per  thousand  observations. 


66  EFFECTS   OF   WINDS   AND    OF 

The  number  of  rejections  was  much  larger  than  one  per  thousand.  As  an 
example,  in  solution  L2,  the  final  barometric  solution  for  Buffalo,  13  days  of 
observation  were  rejected  out  of  a  total  of  243  possible  days  in  the  months 
used  in  the  solution.  This  is  at  the  rate  of  54  observations  per  thousand. 
In  solution  K2,  the  final  barometric  solution  for  Milwaukee,  the  rejections 
were  at  the  rate  of  53  per  thousand. 

The  external  evidence,  obtained  from  a  detailed  study  of  the  separate 
cases,  is  strong  in  indicating  that  as  a  rule,  in  the  rejected  cases,  the  observed 
change  of  elevation  of  the  water  surface  on  the  day  in  question  was  abnormal 
and  due  to  the  first  oscillation  of  a  new  seiche  affecting  the  gage  record  at  the 
station,  or  that  it  was  abnormal  because  of  extremely  rapid  and  irregular 
changes  in  the  barometric  gradients  over  the  lake,  which  departed  widely 
from  the  conditions  postulated  in  the  approximate  theory  used  in  this  in- 
vestigation. Such  rapid  and  irregular  changes  in  barometric  gradients  were 
associated  ordinarily  with  the  passage  of  a  powerful  storm  center,  a  low- 
pressure  area,  over  the  lake  or  near  it.  A  discussion  of  the  seiches  will  be 
found  in  a  later  part  of  this  publication. 

Throughout  the  investigation  any  observation  which  showed  a  residual 
larger  than  3.5  times  the  probable  error  of  the  observation  was  considered 
as  suspicious.  In  each  such  case  a  special  study  of  the  external  evidence 
was  made. 

Early  in  the  investigation  it  was  found  that  such  a  residual  was  frequently 
preceded  or  followed  by  a  residual  of  the  opposite  sign,  which  was  also  of 
considerable  size.  A  study  in  detail  of  many  such  cases,  using  the  evidence 
which  was  external  to  the  least-square  solution,  showed  that  very  frequently 
such  a  case  was  due  to  an  abnormal  effect,  of  one  or  the  other  of  the  two  kinds 
mentioned  above  in  connection  with  rejections,  which  occurred  mainly  within 
the  limits  of  a  single  day.  This  effect  made  the  observed  elevation  of  the 
water  surface  for  that  one  day  either  abnormally  high  or  abnormally  low. 
In  the  first  case  there  was  an  apparently  abnormal  rise  of  the  water  surface 
on  one  day,  followed  on  the  next  day  by  an  abnormal  fall.  In  the  second 
case  there  was  apparently  an  abnormal  fall  on  one  day,  followed  by  .an 
abnormal  rise  on  the  next.  After  many  such  cases  had  been  examined  in 
detail  the  following  rule  as  to  combining  successive  observation  equations 
was  adopted  and  used  thereafter  throughout  the  investigation. 

RULE  FOR  COMBINING  OBSERVATION  EQUATIONS. 

Whenever  any  observation  equation  has  a  residual  larger  than 
3.5  times  the  probable  error  of  a  single  observation,  and  a  residual 
for  an  equation  immediately  preceding  or  immediately  following  in 
time  is  of  the  opposite  sign,  the  two  observation  equations  shall  be 
combined  to  form  one  equation,  provided  the  residual  for  the  new 
combined  observation  equation  will  be  less  than  3.5  times  the 
probable  error  of  a  single  observation. 

Such  a  procedure  rejects  the  observed  elevation  of  the  water  surface  on 
one  day  and  treats  a  two-day  interval  as  the  basis  of  observation  instead  of  a 


BAROMETRIC    PRESSURES   ON   THE    GREAT   LAKES  67 

one-day  interval,  so  far  as  the  one  combined  equation  is  concerned.  It 
retains  all  of  the  observed  facts  as  to  changes  in  barometric  gradients  and 
uses  them  in  the  combined  equation. 

As  already  stated,  page  25,  the  combined  observation  equation  was  formed 
by  adding  the  two  separate  equations  term  by  term. 

If  the  residuals  were  due  entirely  to  accidental  errors,  only  18  equations  per 
thousand  would  have  residuals  larger  than  the  adopted  suspicion  limit,  —3.5 
times  the  probable  error.  Much  less  than  18  equations  per  thousand  should, 
if  the  residuals  were  due  entirely  to  accidental  errors,  be  subject  to  the  com- 
bination rule.  For  if  all  errors  were  accidental,  only  a  few  of  the  suspicious 
residuals  would  be  preceded  or  followed  immediately  by  a  residual  of  oppo- 
site sign  big  enough  to  bring  the  combined  residual  within  the  suspicion 
limit. 

In  table  No.  6  it  is  shown  that  the  number  of  days  of  observation  used  in 
each  solution  exceeded  the  number  of  equations  from  6  per  cent  (at  Mac- 
kinaw) to  22  per  cent  (at  Buffalo).  In  a  few  cases  two  or  more  days  were 
used  in  one  observation  equation,  because  no  record  of  the  elevation  of  the 
water  surface  was  obtainable  from  the  gage  for  one  or  more  days.  In  each 
such  case  an  observation  equation  was  ordinarily  used  covering  the  interval 
during  which  the  gage  record  was  missing.  After  allowing  for  such  cases, 
without  a  definite  count,  it  appears  that  the  number  of  combinations  made 
under  the  above-stated  rule  for  combining  observation  equations  varied 
from  about  50  to  about  200  per  thousand.  This  is  far  in  excess  of  the 
number  of  such  combinations  which  would  occur  if  the  residuals  were  entirely 
due  to  accidental  error.  The  external  evidence  supported  the  combinations, 
of  one  or  the  other  of  the  two  kinds  mentioned  in  connection  with  rejections, 
as  being  justified  by  something  peculiar  to  the  one  abnormal  day. 


DISCREPANCIES  BETWEEN  PAIRS  OF  VALUES. 

Each  least-square  solution  gives  two  values  for  Cw  and  two  values  for  Cn, 
as  noted  on  page  23.  The  discrepancy  between  the  two  values  of  each  pair 
is  a  test  of  the  accuracy  of  the  adopted  value  of  Cw  or  Cn.  Table  No.  14 
gives  the  discrepancies  of  that  character  arising  from  the  final  solutions  for 
barometric  effects  which  gave  the  adopted  barometric  constants  shown  in 
table  No.  6,  page  32.  The  equations  (36)  to  (39),  referred  to  in  the  first 
column,  are  shown  on  page  23. 

According  to  the  laws  of  probability,  the  discrepancy  between  two  values 
should  be  upon  an  average  about  three  times  the  probable  error  of  their  mean. 
In  table  No.  14,  5  of  the  10  discrepancies  are  less  than  10  per  cent  of  the 
larger  of  the  two  C's  for  that  station.  The  extreme  case  is  the  discrepancy 
of  1.94  between  the  two  values  of  Cn  for  Cleveland,  which  is  29  per  cent  of 
the  adopted  value  of  Cn  at  that  station.  Table  No.  14  indicates,  therefore, 
that  the  errors  in  the  computed  values  of  Cw  and  Cn  are  probably  less  than  3 
per  cent  of  the  larger  of  said  values  at  each  station. 


68 


EFFECTS   OF   WINDS   AND    OF 
TABLE  No.  14. 


Buffalo. 

Cleveland. 

Milwaukee. 

Harbor 
Beach. 

Mackinaw. 

Designation  of  solution  
Cw  from  equation  (36)  
Cw  from  equation  (37)  .... 
Discrepancy  between  above 
two  values  
Adopted  mean  of  above  two 
values  

L2 
+  3.74 
+  5.70 

1.96 

+  4.72 

M2 
+1.73 
+2.52 

.79 
+2.12 

K2 
-4.70 
-5.24 

.54 
-4.97 

Nl 

+7.91 
+5.97 

1.94 
+6.94 

Ol 

-2.78 
-2.29 

.49 
-2.54 

Cn  from  equation  (38)  .... 
Cn  from  equation  (39)  
Discrepancy  between  above 
two  values 

-15.65 
-15.60 

.05 

+7.77 
+5.83 

1.94 

+8.33 
+8.56 

.23 

-   .51 
-    .37 

14 

-4.32 
-4.44 

.12 

Adopted  mean  of  above  two 
values                     

-15.62 

+6.80 

+8.44 

-     44 

-4.38 

Each  least-square  solution  gives  two  values  for  the  lag  in  each  barometric 
effect,  E-W  and  N-S,  as  noted  on  page  22.  The  discrepancy  in  each  pair 
between  the  two  computed  values  of  the  lag  is  a  test  of  the  accuracy  of  the 
adopted  value  of  the  lag.  Table  No.  15  gives  the  discrepancies  of  that 
character  arising  from  the  final  solutions  for  barometric  effects.  The  lags 
were  computed  by  use  of  table  No.  2,  as  indicated  on  page  22. 

In  table  No.  15,  of  the  10  possible  discrepancies,  3  did  not  develop,  since 
there  was  a  failure  to  get  a  determination  of  the  lag,  as  indicated  in  the 
footnote. 

Of  the  remaining  7  discrepancies,  5  range  from  2.0  hours  to  7.0  hours,  and 
2  are  0.2  hour  and  0.1  hour,  for  the  lag  in  CB  at  Milwaukee  and  Buffalo, 
respectively. 

Note  that  by  comparison  with  table  No.  7,  page  33,  the  two  pairs 
of  determinations  of  lag  which  gave  the  very  small  discrepancies  noted  at 
Milwaukee  and  Buffalo  were  for  very  large  values  of  Cn,  +8.44  and  —15.62, 
respectively;  also,  that  in  the  case  in  which  both  possible  determinations  of 
lag  failed  for  Cn  at  Harbor  Beach,  the  value  of  Cn  involved  is  very  small, 
-0.44. 

In  general,  then,  it  appears  that  the  lag  in  the  barometric  effects  is 
apparently  determined  within  less  than  an  hour  for  the  largest  effects,  is 
subject  to  a  possible  uncertainty  of  several  hours  for  the  very  smallest 
effects,  and  for  the  effects  of  intermediate  size  the  chances  are  about  even 
that  the  error  in  the  lag  is  only  one  or  two  hours. 

When  one  notes  how  slowly  the  barometric  effects  change  from  hour 
to  hour  it  appears  that  such  determinations  of  lag  represent  a  rather  high 
degree  of  accuracy  in  the  determination  of  barometric  effects.  At  Buffalo, 
where  Cn  is  —15.62,  the  largest  value  of  the  kind  encountered  in  this  in- 
vestigation, the  maximum  change  in  the  N-S  barometric  effect  in  one  hour 
found  in  the  computations  was  0.07  foot.  The  discrepancy  between  the 


BAKOMETRIC    PRESSURES    ON    THE    GREAT   LAKES 


69 


lag  determinations  in  this  case,  0.1  hour,  therefore  corresponded  to  a  dis- 
crepancy in  elevation  of  the  water  surface  of  less  than  0.01  foot. 

TABLE  No.  15. 


Buffalo. 

Cleve- 
land. 

Mil- 
waukee. 

Harbor 
Beach. 

Mack- 
inaw. 

Designation  of  solution  .... 

L2 

M2 

K2 

Nl 

Ol 

Lag  in  Cw  from  
Buii 

-0.5hr 

+5.6hr 

+8.2hr 

+4.4hr 

-S.Ohr 

_             Bwt 
Lag  in  Cw  from  —  —  
Bwo 

* 

+2.1  hr 

+6.2hr 

+8.4hr 

+2.0hr 

Discrepancy  between 

above  two  values  

.... 

3.5hr 

2.0hr 

4.0hr 

7.0hr 

Adopted  mean  of  above 

two  values  

-1      hr 

+4      hr 

+7      hr 

+6      hr 

+2     hr 

Lag  in  Cn  from  

Bni 

+3.3hr 

+0.4hr 

+4.3hr 

* 

l.lhr 

Lag  in  Cn  from  rr-  
Bno 

+3.4hr 

+4.3hr 

+4.1hr 

* 

* 

Discrepancy  between 

above  two  values  

0.1  hr 

3.9hr 

0.2hr 

.... 

Adopted  mean  of  above 

two  values  

+3      hr 

+2      hr 

+4      hr 

+6     hrf 

-1      hr 

For  the  purposes  of  this  table  the  words  hour  and  hours  are  designated  as  hr. 

Values  marked  with  minus  signs  are  negative  lags  or  anticipations. 

*In  these  cases  no  determination  of  the  lag  was  possible,  because  the  two  values  of  which 
the  ratio  should  serve  to  determine  the  lag  were  of  opposite  sign. 

fThis  value,  not  determinable  from  the  observations,  was  assumed  to  be  +6  hours,  the 
same  as  the  lag  as  determined  for  the  E-W  effects  at  this  station. 


STUDY  OF  PROPORTIONALITY  FACTORS. 

Additional  evidence  as  to  the  accuracy  and  reliability  of  the  computed 
barometric  effects  is  afforded  by  a  study  of  the  proportionality  factors  Pv 
and  Pn  at  the  various  stations. 

The  formulas  (16)  and  (17),  page  16,  for  barometric  effects  were  derived 
upon  the  assumption  that  the  water  of  the  lake  remained  continuously  in 
equilibrium  under  the  influence  of  gravity  and  the  barometric  pressure.  The 
proportionality  factors  Pv  and  Pn  were  then  introduced  (see  pages  16-18)  to 
take  account  of  the  modifications  which  would  probably  be  produced  by  fric- 
tion and  by  inertia.  It  was  recognized  that  such  modifications  would  prob- 
ably be  dependent  to  a  considerable  extent  upon  the  configuration  of  the 
shores  and  bottom  of  the  lake  and  might  to  a  considerable  extent  be  peculiar 
to  each  gage  location.  The  modified  equations  for  barometric  effects 
which  were  the  basis  of  this  investigation  are  shown  as  (18)  and  (19)  on 
page  17. 


70 


EFFECTS   OF   WINDS   AND   OF 


Table  No.  16  shows  grouped  together  for  convenient  inspection,  (1)  the 
values  of  Cv  and  Cn  derived  from  the  final  solutions  (see  table  No.  7,  page 
33),  (2)  the  values  of  Rw  and  Rn  computed  as  indicated  in  formula  (17), 
page  16,  for  each  gage  station,  and  (3)  the  various  values  of  Pw  and  Pn  de- 
rived by  substitution  of  the  values  of  Cw,  Cn,  Rw,  and  Rn  in  equation  (19). 

TABLE  No.  16. 


Buffalo. 

Cleveland. 

Milwaukee. 

Harbor 
Beach. 

Mackinaw. 

Designation    of    solu- 
tion 

L2 

M2 

K2 

Nl 

Ol 

C«  
Rw                .... 

+  4.72 
+  3.55 

+2.12 
-   .75 

-4.97 
-5.14 

+6.94 
+3.10 

-  2.54 
-     .12 

Pw 

-f  1  33 

—2  83 

+     97 

+2  24 

+21  17 

Cn  

Rn 

-15.62 
—  1  72 

+6.80 
+1  39 

+8.44 
+3  20 

-0.44 
+1  38 

-  4.38 
—  3  03 

Pn  

+  9.08 

+4.89 

+2.64 

-   .32 

+  1.45 

If  the  actual  barometric  effects  were  those  corresponding  to  continuous 
equilibrium  of  the  water  under  the  influence  of  gravity  and  barometric 
pressure,  and  if  all  assumptions  made  in  deriving  formulse  (16)  and  (17)  were 
true,  the  computed  values  of  Pw  and  Pn  would  be  1.00  at  every  station 
within  the  limits  fixed  by  the  errors  of  observation.  The  wide  range  of 
values  of  Pw  and  Pn  in  table  No.  16,  from  —2.83  to  +21.17,  is  ample  evidence 
that  decided  modifications  of  the  barometric  effects  are  produced  by  friction 
and  inertia,  and  that  there  are  probably  appreciable  errors  in  the  assumptions 
used  as  a  basis  for  the  computations.  The  wide  range  also  indicates  that 
the  modifications  and  errors  are  peculiar  to  each  station,  not  common  to 
them  all. 

The  extreme  value  +21.17  for  Pw  at  Mackinaw  is  probably  due  to  two 
causes.  The  gage  at  Mackinaw  is  very  slightly  to  the  westward  of  the 
center  of  gravity  of  the  area  of  Lake  Michigan-Huron  (see  plate  2).  Hence, 
the  value  of  Rv  for  that  gage  is  very  small,  —0.12.  The  Strait  of  Mackinac 
probably  acts  as  a  throttle,  to  a  certain  extent,  between  Lake  Michigan  and 
Lake  Huron.  It  probably  tends  to  delay  the  delivery  of  water  back  and 
forth  between  the  two  lakes,  which  must  occur  under  the  influence  of  baro- 
metric changes  if  the  two  lakes  are  to  act  as  one.  Any  delay  in  the  delivery 
of  the  water  through  the  Strait  of  Mackinac  tends  to  make  the  two  lakes  act 
temporarily  as  separate  lakes.  If  the  water  surface  at  Mackinaw  gage 
acted  as  if  it  were  a  part  of  the  surface  of  Lake  Huron  alone,  the  Lw  of  such 
a  formula  as  (17)  for  that  gage  should  be  measured  from  the  center  of 
gravity  of  Lake  Huron  alone  and  would  be  much  greater  than  the  Lw  which 
was  actually  measured  from  the  center  of  gravity  of  the  combined  lakes. 
In  that  case,  Rw  would  be  much  larger  than  —0.12,  while  Pw  would  be  much 
smaller  than  +21.17.  In  other  words,  if  the  oscillation  of  the  water  of  Lake 


BAROMETRIC   PRESSURES   ON   THE    GREAT   LAKES  71 

Huron  under  the  influence  of  barometric  changes  is  about  its  own  center  of 
gravity  as  a  nodal  point,  the  rise  and  fall  at  Mackinaw  gage  would  be  much 
greater  under  fluctuation  of  E-W  barometric  gradients  than  if  the  oscilla- 
tion were  about  the  center  of  gravity  of  the  combined  lakes — Lake  Michigan- 
Huron — as  assumed  in  the  computations.  The  extremely  large  value  of 
Pv  at  the  Mackinaw  gage,  +21.17,  is  believed,  therefore,  to  be  a  strong 
indication  of  a  throttling  effect  in  the  Strait  of  Mackinac. 

The  negative  value  of  Pn  at  Harbor  Beach,  —0.32,  is  believed  to  be  due  to 
a  peculiar  effect  of  the  Strait  of  Mackinac  arising  from  the  fact  that  it  is  the 
only  connection  between  Lake  Michigan  and  Lake  Huron,  and  that  said 
connection  is  almost  at  the  extreme  northern  end  of  each  lake.  The  fact 
that  Bn0  and  Bni  at  Harbor  Beach  (see  table  No.  6,  page  32)  have  apparently 
abnormal  negative  signs,  while  Bn2  and  B&  have  the  positive  signs,  which 
would  be  normal  according  to  the  derivation  of  the  formula  used  in  this  in- 
vestigation, is  also  believed  to  be  due  to  the  same  cause. 

No  adequate  explanation  is  offered  for  the  apparently  abnormal  value  of 
Pw  at  Cleveland,  —2.83.  It  is  surmised  that  it  is  related  in  some  way  to  the 
following  three  facts:  (1)  the  center  of  gravity  of  Lake  Erie,  which  normally 
should  be  the  nodal  point  for  barometric  effects,  is  not  far  to  the  eastward  of 
Cleveland;  (2)  the  nodal  line  for  east  or  west  winds  on  Lake  Erie  is  to  the 
westward  of  Cleveland ;  and  (3)  there  is  a  decided  bend  in  the  south  shore  of 
Lake  Erie  at  Cleveland.  Note  that  the  minus  sign  for  Pw  at  Cleveland 
indicates  that  the  actual  nodal  point  for  barometric  effects  on  Lake  Erie  is 
to  the  westward  of  Cleveland. 

The  remaining  seven  values  of  Pw  and  Pn  on  which  no  special  comments 
have  been  made  are  all  positive,  their  mean  value  is  +3.23,  and  the  largest 
two  values  are  +9.08  and  +4.89  for  Pn  at  Buffalo  and  Cleveland,  respec- 
tively. Inertia  tends  strongly  to  make  Pw  and  Pn  greater  than  +1.00. 
Other  evidence,  set  forth  later  in  this  publication,  indicates  that  inertia 
effects  upon  the  elevation  of  the  water  surface  are  probably  much  greater 
on  Lake  Erie  than  on  Lake  Michigan-Huron.  The  effects  of  errors  in 
assumption  No.  2  (see  page  16)  are  probably  greater  for  N-S  effects  on  Lake 
Erie  than  in  any  other  case  in  this  investigation,  since  the  ratio  of  the  dis- 
tance from  point  5  to  point  7  (see  plate  2)  to  the  maximum  extent  of  Lake 
Erie  in  the  N-S  direction  is  greater  than  any  other  similar  ratio  involved  in 
assumption  No.  2.  Note  that  for  the  reason  stated  on  page  16  the  de- 
parture of  the  facts  from  assumption  No.  2  is  likely  to  be  such  as  to  make  the 
values  of  Pv  and  Pn  considerably  greater  than  +1.00.  For  the  reasons 
indicated  briefly  in  this  paragraph,  it  is  believed  that  the  seven  values  of  Pw 
and  Pn  here  commented  upon  have  in  them  no  indications  of  inaccuracy  of 
observation  or  computation  or  unreliability  of  the  theory  on  which  the 
computations  were  made. 

Taking  into  account  all  ten  values  of  Pw  and  Pn,  the  comments  made 
upon  them  above,  and  other  consideration  of  details  not  here  set  forth,  the 
general  conclusions  reached  are  as  follows: 


72  EFFECTS   OF   WINDS   AND    OF 

(1)  The  barometric  effects  as  computed  have  in  general,  barring  certain 
classes  of  exceptional  cases  referred  to  in  (4),  the  degree  of  accuracy  indicated 
by  the  computed  probable  errors. 

(2)  There  are  decided  modifications  of  the  barometric  effects,  due  to  the 
configuration  of  the  shores  and  bottom  of  the  lake,  which  modifications  are 
peculiar  to  each  gage  station  and  in  general  increase  the  magnitude  of  the 
effects. 

(3)  The  errors  in  assumption  No.  2  probably  contribute  to  making  Pw 
and  Pn  greater  than  +1-00. 

(4)  At  times,  when  the  changes  in  barometric  gradients  are  unusually  rapid 
and  irregular,  such  as  the  times  when  a  well-developed  low-pressure  area  is 
passing  over  or  near  a  lake,  the  errors  in  the  computed  barometric  effects  are 
probably  abnormally  large.     At  such  a  time  the  shape  of  the  wave  produced 
on  the  lake  surface  by  barometric  changes  is  unusual,  and  therefore  one  may 
expect  the  modifications  in  it  due  to  inertia  and  dependent  upon  the  con- 
figuration of  the  shores  and  bot.tom  to  be  unusual. 

CONCLUSIONS  ON  ACCURACY  OF  COMPUTED  BAROMETRIC   EFFECTS. 

A  comparison  of  each  final  least-square  solution,  such  as  L2  at  Buffalo, 
with  earlier  least-square  solutions,  furnishes  very  valuable  and  instructive 
evidence  as  to  the  accuracy  and  reliability  of  the  conclusions  from  the  final 
solutions.  But  such  evidence  is  made  up  of  many  details  and  many  con- 
siderations of  such  a  character  that  they  can  not  be  briefly  presented.  Hence, 
it  must  suffice  here  to  state  that  such  evidence  furnished  strong  corrobo- 
ration  of  the  foregoing  discussion  of  the  errors  in  the  computed  barometric 
effects. 

The  general  conclusions,  based  upon  all  the  evidence  available,  which 
have  been  reached  in  regard  to  the  computed  barometric  effects  at  the  five 
stations  Buffalo,  Cleveland,  Milwaukee,  Harbor  Beach,  and  Mackinaw  are: 

(1)  The  errors  of  the  computed  daily  barometric  effects  are  probably  less 
than  5  per  cent  of  said  effects  in  about  one-half  of  all  days. 

(2)  On  a  small  percentage  of  exceptional  days,  at  times  when  the  baro- 
metric gradients  are  changing  rapidly  and  irregularly,  usually  when  a  well- 
developed  low-pressure  area  is  over  or  near  the  lake,  the  computed  baromet- 
ric effects  are  subject  to  abnormally  large  errors,  much  greater  than  5  per 
cent.     These  exceptional  days  are  usually  solitary,  not  in  groups. 

ACCURACY  OF  COMPUTED  WIND  EFFECTS. 

As  shown  on  page  63,  the  final  value  adopted  for  the  constant  Cx  of  the 
fundamental  formula  for  wind  effects — (51)  on  page  39 —  was  +0.088,  which 
gave  as  the  definite  numerical  formula  for  wind  effects  (69)  on  page  63. 

What  is  the  accuracy  of  the  value  +0.088?  This  value  is  a  weighted 
mean  of  four  separate  values  from  four  separate  least-square  solutions,  as 
shown  in  table  No.  13,  page  63.  Its  probable  error  is  ±0.006,  as  there. 


BAROMETRIC    PRESSURES   ON   THE    GREAT   LAKES  73 

shown,  computed  rigorously  from  the  normal  equations  and  the  residuals  of 
the  four  solutions.  If  all  errors  affecting  the  value  +0.088  were  of  the  ac- 
cidental class,  the  chances  would  therefore  be  even  that  said  value  is  correct 


... .     1         ,  /0.006      1 

within  —  part  ( =  — 

15          V0.088     15 


The  four  separate  least-square  solutions  are,  as  indicated  in  table  No.  12, 
page  62,  based  on  four  separate  and  independent  sets  of  observed  data, 
elevations  of  the  water  surface  at  two  different  gages — Buffalo  and  Cleveland 
— and  including  2,291  hours  of  observation  scattered  over  100  days.  If 
any  systematic  or  constant  errors  affected  the  computed  values  of  Cx  from 
the  separate  solutions,  they  would  be  apt  to  appear  in  the  comparison  of 
those  four  values  which  is  shown  in  table  No.  13.  If  the  errors  are  all  of  the 
accidental  class,  each  of  such  residuals  as  are  shown  in  the  last  column  of 
that  table  should  have  an  even  chance  of  being  less  than  the  probable  error 
written  in  the  same  line  in  that  table.  Note  that  the  residual  for  solution 
W29,  +0.009,  is  much  less  than  the  corresponding  probable  error,  ±0.0175; 
that  the  residual  for  solution  W26,  +0.016,  is  very  slightly  greater  than  the 
corresponding  probable  error,  ±0.0142;  and  that  the  residual  which  is  largest 
in  proportion  to  its  corresponding  error  is  that  for  solution  W25,  —0.017, 
which  is  1.9  times  the  corresponding  probable  error,  ±0.0087.  According 
to  the  laws  of  probability,  such  a  residual,  1.9  times  the  probable  error, 
should  occur  on  an  average  once  in  five  times.  It  appears,  then,  that  the 
agreement  of  the  four  values  of  Cx  from  the  four  solutions  is  so  close  as  to 
furnish  no  indication  of  systematic  or  constant  errors.  The  degree  of 
agreement  indicates  that  all  errors  are  of  the  accidental  class,  and  therefore 
that  the  degree  of  accuracy  of  the  value  +0.088  is  that  indicated  by  its 
computed  probable  error,  ±0.006. 

The  same  rules  as  to  rejections  and  combinations  were  used  in  the  com- 
putation of  wind  effects  as  have  already  been  stated  in  connection  with 
the  computation  of  barometric  effects  (see  pages  65-66). 

The  same  considerations  which  led  there  to  the  conclusion  that  there  was 
no  danger  that  a  fictitious  accuracy  had  been  imparted  to  the  computed 
barometric  effects  as  a  result  of  the  rejections  and  combinations  also  led  to 
the  same  conclusion  with  respect  to  the  computed  wind  effects. 

As  shown  in  table  No.  12,  of  the  total  of  2,377  hours  of  observation  avail- 
able, 2,291  hours  were  used.  That  is,  only  86  hours,  or  less  than  4  per  cent, 
were  rejected. 

The  total  number  of  separate  observation  equations  was  2,173 — 118  less, 
or  5  per  cent  less,  than  the  total  number  of  hours  used.  That  is,  the  combi- 
nations of  equations  which  were  made  were  such  as  to  combine  5  per  cent 
of  the  hours  with  adjacent  hours  instead  of  using  them  in  separate  inde- 
pendent equations. 

In  addition  to  the  rejections  and  combinations  made  under  the  regular 
rules,  referred  to  above,  in  solution  W29  at  Cleveland,  a  few  other  combi- 
nations and  rejections  were  made  on  the  basis  of  evidence  external  to  the 


74  EFFECTS   OF  WINDS   AND   OF 

solution.  These  were  based  mainly  upon  evidence  which  indicated  that 
when  there  was  an  apparent  shift  of  the  wind  direction  between  the  direc- 
tions N1  and  NE  or  between  the  directions  S  and  SW  the  water  surface  at 
Cleveland  in  frequent  cases  did  not  respond  in  the  normal  manner.  There 
were  two  surmises  as  to  the  reason,  one  connected  with  the  possible  be- 
havior of  the  water  itself  and  one  a  suspicion  as  to  the  wind  record. 

Plate  2  shows  that  with  a  shift  of  the  wind  from  N  to  NE  or  from  S  to 
SW  there  is  a  very  large  shift  in  the  position  of  the  nodal  line  on  Lake  Erie. 
This  means  that  for  a  slight  change  in  direction  of  the  wind,  only  45°,  there 
must  be  a  transfer  of  an  unusually  large  amount  of  water  over  an  unusually 
long  distance  before  the  new  steady  regime  is  established.  Accordingly,  at 
the  time  of  said  shifts  it  is  possible  that  there  are  decidedly  unusual  tem- 
porary fluctuations  in  elevations  of  water  surface  at  various  points  due  to 
inertia  effects  in  the  unusual  currents  which  must  occur  at  those  times.  Cleve- 
land, lying  between  the  two  locations  of  the  nodal  line  for  S  and  for  SW 
winds,  may  be  especially  subject  to  such  temporary  fluctuations  of  elevation 
of  water  surface. 

The  other  surmise  arose  from  two  considerations: 

First,  in  talking  with  officials  of  the  Weather  Bureau  in  both  Chicago  and 
Washington  the  idea  was  several  times  advanced  that  at  particular  stations 
there  might  be  errors  in  the  recorded  direction  of  the  wind  at  the  station, 
due  to  the  immediate  surroundings  of  the  station,  such  that  the  recorded 
wind  direction  of  the  station  might  not  be  truly  representative  of  the  wind 
direction  in  a  large  region  around  the  station.  For  example,  it  is  possible 
that  for  certain  hours  a  wind  may  be  recorded  as  NE  at  the  Weather  Bureau 
station  at  Cleveland  though  the  actual  wind  blowing  at  that  time  over  the 
western  half  of  Lake  Erie  and  influencing  the  water  surface  might  be  a  north 
wind  rather  than  a  northeast  wind. 

Second,  for  certain  days  in  which  there  seemed  to  be  reason  to  suspect  the 
validity  of  the  Cleveland  wind  directions  in  the  respect  just  noted,  a  com- 
parison was  made  of  the  wind  directions  recorded  at  Cleveland  and  those 
recorded  at  Buffalo  and  other  stations.  These  comparisons  in  some  cases 
indicated  fickle  fluctuations  of  wind  between  N  and  NE  as  recorded  at  Cleve- 
land when  the  record  at  other  stations  showed  no  reason  to  apprehend  such 
fluctuations.  It  was  therefore  surmised  that  possibly  there  might  be  a  tend- 
ency to  station  error  in  the  wind  direction  at  Cleveland,  affecting  especially 
the  directions  N,  NE,  S,  and  SW.  This  is  the  only  case  in  this  investigation 
in  which  there  appears  to  be  any  reason  to  suspect  an  appreciable  station 
error  in  either  wind  direction  or  wind  velocity. 

The  total  number  of  rejections  and  combinations  made  on  the  unusual 
basis  indicated  above  was  moderate.  Table  No.  12  shows  that  the  total 
number  of  rejections  in  solution  W29  at  Cleveland  was  only  7  per  cent  and 
the  total  of  combinations  was  only  6  per  cent,  whereas  the  corresponding 
percentages  for  all  four  solutions  combined  were  4  and  5,  respectively. 

The  external  evidence  seemed  to  be  ample  and  convincing  that  all  these 


BAEOMETRIC   PRESSURES   ON   THE    GREAT   LAKES 


75 


rejections  and  combinations  should  be  made.  Aside  from  the  special  cases 
in  solution  W29  at  Cleveland  indicated  above,  the  rejections  and  combina- 
tions appear  to  be  due,  as  a  rule,  to  seiches,  and  especially  to  the  first 
and  largest  oscillation  of  a  new  seiche  which  has  just  been  started  by  an 
unusually  vigorous  impulse  given  by  a  change  in  wind  or  by  a  change  in 
barometric  gradients. 

Aside  from  the  four  final  solutions  for  wind  effects  of  which  the  results  are 
given  in  table  No.  13,  page  63,  and  which  fixed  the  adopted  value  of  Cx, 
three  other  wind  solutions  for  different  gage  stations  were  also  made, 
namely,  one  each  at  Milwaukee,  Harbor  Beach,  and  Mackinaw.  These 
three  gave  determinations  of  Cx  which  have  such  large  probable  errors  that 
the  values  were  not  used  in  fixing  the  adopted  value  of  Cx.  These  solutions 
were  not  made  with  all  the  refinements  of  the  final  solutions.  Nevertheless, 
they  furnish  a  valuable  check  on  the  correctness  of  the  theory  as  to  wind 
effects  which  has  been  used  in  the  investigation. 

In  table  No.  17,  showing  the  results  of  these  three  solutions,  the  assigned 
weights  are  on  the  same  basis  as  the  weights  shown  in  table  No.  13. 

TABLE  No.  17. 


Probable 
error  of 

Cx. 

Assigned 
weight. 

Cx 

=  Cp+Ca. 

Residual  from 
adopted  final 
value  of 
Cx,  viz.,  +.088. 

Solution  W16,  Milwaukee.  .  . 
Solution  W17,  Harbor  Beach 
Solution  W18,  Mackinaw.  .  . 

Sum  of  weights  

±0.134 
±   .081 
±   .245 

0.06 
.15 
.02 

-0.229 
-   .218 
+   .246 

+0.317 
+   .306 
-   .158 

=    .23 

Note  that  the  sum  of  the  weights  for  these  three  solutions  is  less  than 
0.01  as  great  as  the  sum  of  the  weights  for  the  four  final  solutions,  as  shown 
in  table  No.  13. 

Note  that  the  residuals  for  these  three  solutions  are  not  large  enough  to 
prove  that  systematic  or  constant  errors  are  present.  One  of  the  three 
residuals  is  less  than  the  corresponding  probable  error,  and  in  the  extreme 
case,  solution  W17,  the  residual  is  3.8  times  the  corresponding  probable 
error.  According  to  the  laws  of  probability,  a  residual  3.8  times  the  cor- 
responding error  should  occur  once  in  about  100  times. 

If  these  three  solutions,  with  their  proper  weights,  as  shown  above,  were 
used  with  the  four  solutions  of  table  No.  13  to  fix  the  final  adopted  value  of 
Cx,  that  value  would  be  +0.086,  differing  only  0.002  from  that  actually 
adopted. 

Consider  the  values  of  2^  which  are  shown  in  table  No.  10,  page  53,  and 
note  the  contrast  between  the  values  for  the  two  lakes  concerned.  The  maxi- 
mum value  for  the  Lake  Erie  stations  is  8.32  and  the  minimum  value  is  1.72. 


76 


EFFECTS   OF   WINDS   AND    OF 


On  the  other  hand,  the  maximum  value  at  any  of  the  three  Lake  Michigan- 
Huron  stations  is  0.95,  but  little  more  than  one-half  of  the  minimum  on 
Lake  Erie.  Of  the  24  values  at  the  Lake  Michigan-Huron  stations,  6  are 
less  than  one-tenth  as  large  as  the  minimum  (1.72)  at  Lake  Erie  stations. 
Roughly  speaking,  then,  the  Lake  Michigan-Huron  effects,  which  must  be 
proportional  to  2X  if  the  theory  used  in  this  investigation  is  correct,  must  be 
less  than  one-tenth  as  large  as  the  wind  effects  on  Lake  Erie,  at  the  stations 
under  consideration.  Such  minute  wind  effects  as  those  postulated  on 
Lake  Michigan-Huron  must  be  very  difficult  to  detect  by  the  observation  of 
water-surface  elevations,  masked  as  they  necessarily  are  by  the  much  larger 
barometric  effects  and  by  seiches. 

Table  No.  18  shows  how  small  the  wind  effects  are  at  the  three  Lake 
Michigan-Huron  stations,  both  in  absolute  units  and  in  comparison  with 
the  wind  effects  at  the  two  stations  on  Lake  Erie.  The  values  are  as 
computed  from  formula  (69),  page  63. 

TABLE  No.  18 — Maximum  value  of  the  wind  effect  at  any  hour  within  the  limits  of  this 
investigation  at  various  gage  stations. 


Date  and  hour. 

Wind  veloc- 
ity in  miles 
per  hour. 

Wind 
direction. 

Wind 
effect, 
feet. 

Buffalo 

Oct  27  1910  10  a  m 

53 

SW 

1  003 

Cleveland  
Milwaukee  
Harbor  Beach.  .  . 
Mackinaw.  . 

Oct.  27,  1910,  7  a.m. 
Sept.  6,  1911,  midnight 
July  24,  1911,  3  p.m. 
Aug  25  1910  6pm 

41 
31 
35 
33 

W 

E 
W 
W 

.186 
.032 
.034 
016 

When  it  is  realized  that  the  maximum  wind  effect  at  any  one  of  the  three 
stations  on  Lake  Michigan-Huron  during  the  months  covered  by  the  in- 
vestigation was  only  0.034  foot  even  for  a  single  hour,  as  shown  in  the  table, 
and  that  during  more  than  one-half  of  the  time  the  wind  effect  is  certainly 
less  than  0.010  foot,  then  it  is  clear  that  the  large  probable  errors  in  table 
No.  17  are  due  to  this  cause.  So,  also,  are  the  abnormal  minus  signs  on  two 
of  the  computed  values  of  Cz.  The  wind  effects  were  too  small  to  show 
through  the  mask  of  barometric  effects  and  seiches. 

To  what  extent  is  the  accuracy  of  the  computed  wind  effects  reduced  by 
uncertainty  as  to  the  true  value  of  the  exponent  of  h  in  formula  (51),  page 
39?  It  has  already  been  stated,  pages  54-55,  that  no  theory  is  known 
which  is  deemed  adequate  to  fix  the  value  of  this  exponent,  that  it  has  in  this 
investigation  been  derived  from  the  observations,  that  the  observations 
indicate  2.4  to  be  the  most  probable  value,  and  that  the  belief  has  been 
reached  that  whatever  error  exists  in  this  adopted  value  has  a  very  slight 
influence  on  the  accuracy  of  the  computed  wind  effects.  It  is  appropriate 
to  indicate  somewhat  more  definitely  at  this  point  the  basis  of  the  belief 
stated  in  the  last  part  of  the  preceding  sentence. 


BAROMETRIC   PRESSURES   ON   THE    GREAT   LAKES  77 

At  Cleveland,  six  least-square  solutions  for  wind  effects  were  made,  in 
which  the  assumed  values  of  the  exponent  were  2.0,  2.2,  2.3,  and  2.4.  Com- 
parisons were  made  between  these  solutions,  in  pairs,  to  determine  as  far  as 
feasible  the  effect  of  the  assumed  change  of  exponent  between  the  two 
solutions  of  each  pair.  The  change  of  exponent  from  2.0  to  2.2  reduced  Sw1 
by  3  per  cent,  and  other  kinds  of  internal  evidence  showed  clearly  that  2.2 
is  nearer  the  truth  than  2.0.  The  expression  Sy2  is  used  to  indicate  the  sum 
of  the  squares  of  the  residuals  in  a  solution.  It  is  evident  that  the  closer 
the  approach  to  the  truth  in  all  assumptions  the  smaller  will  be  Sv2.  The 
change  of  the  assumed  exponent  from  2.2  to  2.3  reduced  ?,v*  only  1  per 
cent,  and  from  2.3  to  2.4  reduced  it  again  only  1  per  cent.  In  these  last 
two  steps  the  other  evidence  than  that  given  by  Sv2  was  not  clear. 

At  Buffalo,  seven  least-square  solutions  for  wind  effects  were  made,  in 
which  the  assumed  values  of  the  exponent  were  2.0,  2.3,  2.4,  and  2.5.  There 
was  decisive  evidence  that  the  change  from  2.0  to  2.3  was  an  approach  to  the 
truth.  But  among  the  six  of  the  seven  solutions  which  were  based  on  the 
assumed  values  2.3,  2.4,  and  2.5  for  the  exponent,  the  apparent  change  in  Sv* 
due  to  change  of  exponent  over  this  range  of  0.2  was  only  1  per  cent  or  less. 
Other  evidence  was  not  clear  in  favor  of  any  choice  between  2.4  and  2.5. 

Hence,  the  conclusions  reached  from  a  consideration  of  the  evidence  as  a 
whole  are  (a)  that  the  most  probable  value  of  the  exponent  is  2.4,  (6)  that 
the  true  value  may  be  as  low  as  2.3  or  as  high  as  2.5,  and  (c)  that  the  error 
in  the  assumed  value  2.4  probably  produces  a  very  small  otherwise-avoidable 
error  in  the  final  computed  daily  wind  effects,  provided  the  one  value  of  the 
exponent  is  carried  consistently  throughout  the  whole  computation. 

The  fact  that  a  change  of  0.1  in  the  exponent  either  way  from  2.4  pro- 
duces a  change  of  only  1  per  cent  or  less  in  Sz;2  is  the  main  basis  for  conclusion 
(c)  above,  which  is  applicable  primarily  to  daily  wind  effects.  Such  a  daily 
effect  is  the  mean  of  24  hourly  effects,  which  usually  involve  winds  of  a  con- 
siderable range  of  velocity  and  usually  a  change  of  direction.  It  is  conceded 
that  the  computed  wind  effect  for  the  hour  of  maximum  velocity  may  be 
appreciably  in  error  on  account  of  the  error  in  the  exponent,  but  the  other 
values  for  lighter  winds  during  the  day  will  tend  normally  to  have  much  less 
error  from  this  cause,  and  for  the  extremely  light  winds  the  error  tends  to  be 
reversed  in  sign.  The  net  result  is  a  daily  wind  effect  based  on  24  hourly 
values  having  little  error  due  to  the  cause  in  question. 

The  adopted  exponent,  2.4,  depends  on  deductions  from  observations, 
not  primarily  on  theory,  and  is  believed  to  be  as  good  an  approximation  to 
the  truth  as  is  needed  for  the  prime  purpose  of  this  investigation. 

The  evidence  as  set  forth  on  pages  72-77,  mainly  from  the  four  final  least- 
square  solutions  for  wind  effects,  is  abundantly  corroborated  by  the  many 
earlier  least-square  solutions  made  in  series  on  the  general  plan  indicated 
on  pages  6-8.  There  were  in  all  29  least-square  solutions  devoted  en- 
tirely to  a  study  of  wind  effects,  and  37  other  solutions  in  which  the  wind 
effects  were  studied  in  conjunction  with  other  matters. 


78  EFFECTS   OF   WINDS   AND    OF 

ACCURACY  OF  CORRECTED  ELEVATIONS  OF  WATER 
SURFACE. 

There  has  been  presented  in  some  detail  in  the  preceding  pages  the 
manner  in  which  the  barometric  effects  and  wind  effects  have  been  com- 
puted for  five  gage  stations  on  the  Great  Lakes.  Some  of  the  evidence  has 
been  set  forth  as  to  the  accuracy  of  these  computed  values.  It  is  important 
to  secure  as  decisive  tests  as  are  feasible  of  the  conclusions  which  have  been 
reached  as  to  the  accuracy  and  reliability  of  the  corrections  for  barometric 
effects  and  wind  effects.  With  that  end  in  view,  the  tabulations  and  com- 
ments of  the  following  pages  dealing  with  observed  and  corrected  elevations 
of  the  water  surface  for  each  day,  for  each  interval  of  five  days,  and  for 
intervals  of  one  month  and  one  season  are  set  forth  in  turn  for  each  of  the 
five  gage  stations  and  for  each  of  the  two  lakes  considered  as  a  unit.  It  is 
believed  that  these  tabulations  and  comments  furnish  the  most  decisive 
tests  of  accuracy  and  reliability  that  are  feasible  within  the  time  limits  of  this 
investigation  and  of  its  exposition  in  print. 

In  tables  Nos.  19-23,  which  follow  in  order  for  Buffalo,  Cleveland,  Mil- 
waukee, Harbor  Beach,  and  Mackinaw,  the  barometric  corrections  and  the 
wind  corrections  as  shown  were  computed  as  indicated  on  pages  36-39, 
63-64.  These  are  corrections,  and  are  therefore  of  the  opposite  sign  from 
barometric  effects  and  wind  effects.  The  observed  elevation  for  each  day 
as  shown  is  the  mean  of  24  hourly  elevations  of  the  water  surface  as  ob- 
served at  the  gage  specified.  The  corrections  for  barometric  effect  and 
wind  effect  for  the  day  being  applied  to  the  observed  elevation  gives  the 
corrected  elevation  as  shown  in  the  next  column.  This  corrected  eleva- 
tion is  the  best  value  of  the  mean  elevation  of  the  whole  lake  surface 
that  can  be  obtained  from  that  gage  for  that  day,  as  distinguished  from 
the  elevation  of  any  part  of  the  surface.  The  main  purpose  of  these  tables, 
though  not  the  only  one,  is  to  show  the  degree  of  accuracy  and  reliability 
with  which  the  mean  elevation  of  the  whole  lake  surface  is  determined 
from  day  to  day,  and  thereby  the  fluctuation  in  total  content  of  the  lake 
determined. 

All  elevations  in  these  tables  are  referred  to  mean  sea-level. 

In  the  column  headed  "5-day  observed  mean,"  each  value  is  in  nearly  all 
cases  the  mean  of  five  daily  observed  elevations.  The  exceptions  are  of  two 
kinds.  First,  in  some  cases  the  group  includes  six  values,  and  the  mean  is 
therefore  for  six  days.  For  example,  the  six  days,  July  26-31,  1910,  are 
grouped  together  in  order  to  have  the  beginning  of  the  next  5-day  group  on 
the  first  day  of  the  month.  Second,  the  group  is  sometimes  for  less  than 
five  days  when  there  has  been  an  interruption  in  the  gage  record.  For 
example,  there  was  no  observed  elevation  available  for  August  11,  1910. 
Hence,  the  group  normally  covering  the  days  August  11-15  contains  but  four 
values,  and  the  mean  in  the  column  marked  "5-day  observed  mean"  is  for 
four  values  only. 


BAROMETRIC   PRESSURES   ON   THE   GREAT  LAKES  79 

The  column  marked  "5-day  corrected  mean"  bears  the  same  relation 
to  the  corrected  elevations  as  the  column  marked  "5-day  observed  mean" 
does  to  the  observed  elevations.  The  explanation  made  in  the  preceding 
paragraph  for  groups  containing  six  values  or  less  than  five  values  also 
applies  to  the  corrected  elevations  and  5-day  corrected  means. 

In  the  column  "Corrected  elevation,"  an  occasional  value  is  inclosed  in 
parentheses  and  the  corresponding  residual  is  also  inclosed  in  parentheses. 
Each  corrected  elevation  contained  in  a  parenthesis,  such,  for  example,  as 
2.36  on  June  2, 1910,  at  Buffalo,  is  one  which  has  been  identified  by  a  definite 
criterion,  based  on  the  preceding  investigation,  as  being  an  abnormal  dis- 
turbed value  which  should  be  rejected  in  taking  means.  It  is  so  re- 
jected in  this  tabulation.  The  criterion  will  be  stated  later  in  the  proper 
context. 

The  residuals  of  the  daily  observed  elevations  from  the  5-day  observed 
means,  and  the  corresponding  residuals  of  daily  corrected  elevations  from  the 
5-day  corrected  means,  as  shown  side  by  side  in  the  last  two  columns  of  the 
table,  furnish  an  instructive  indication  of  the  relative  accuracy  of  the 
observed  and  corrected  elevations. 

The  preceding  general  explanations  also  apply  to  tables  Nos.  20  to  23. 

MEAN  ELEVATIONS  OF  LAKE  ERIE. 

In  tables  Nos.  19  and  20  there  are  given  elevations  of  the  water  surface  of 
Lake  Erie  as  observed  at  the  Buffalo  gage  and  at  the  Cleveland  gage. 
Before  this  investigation  had  been  made,  the  best  available  approximation 
to  the  mean  elevation  of  the  whole  surface  of  Lake  Erie  on  a  given  day 
would  have  been  assumed  to  be  the  mean  of  these  two  observed  elevations 
at  the  two  gages.  Also,  there  are  given  in  tables  Nos.  19  and  20  the  cor- 
rected elevations  from  those  gages.  Each  such  corrected  elevation  is  a 
value  for  the  elevation  of  the  mean  surface  of  Lake  Erie  after  the  cor- 
rections have  been  applied  for  disturbances  at  each  gage  by  barometric  pres- 
sures and  by  winds.  The  mean  of  these  two  corrected  elevations,  one  for 
the  Buffalo  gage  and  one  for  the  Cleveland  gage,  is  recognized  from  this 
Investigation  to  be  the  best  approximation,  on  any  given  day,  to  the  mean 
elevation  of  the  whole  surface  of  Lake  Erie  for  that  day. 

Table  No.  24  serves  to  place  in  juxtaposition  these  two  mean  values  of 
the  mean  elevation  of  the  whole  surface  of  Lake  Erie,  one  without  corrections 
and  the  other  with  corrections  for  barometric  effects  and  wind  effects.  The 
table  serves  to  enable  one  to  compare  the  two  sets  of  values,  study  their  ac- 
curacy by  means  of  the  5-day  means  and  the  residuals,  and  so  make  prog- 
ress in  testing  the  over-all  accuracy  and  reliability  of  the  barometric  cor- 
rections and  wind  corrections. 

In  table  No.  24  the  observed  elevation  for  any  day  was  obtained  by 
merely  taking  a  mean  of  the  two  observed  elevations  for  that  day  as  re- 
corded in  table  No.  19  for  Buffalo  and  table  No.  20  for  Cleveland. 


80 


EFFECTS   OF   WINDS   AND    OF 


TABLE  No.  19 — Observed  and  corrected  elevations  of  water  surface  at  the  Buffalo  Gage  on 

Lake  Erie. 


Date. 

Baro- 
metric 
cor- 
rection. 

Wind 
cor- 
rection. 

Observed 
eleva- 
tion 
570  +. 

Cor- 
rected 
eleva- 
tion 
570  +. 

5-day 
observed 
mean 
570  +. 

5-day 
corrected 
mean 
570  +. 

Residuals  from 
5-day  means. 

Obs. 

Cor. 

1910. 

feet. 

feet. 

feet. 

feet. 

feet. 

feet. 

feet. 

feet. 

June    1... 

-0.31 

-0.11 

2.94 

2.52 

-0.38 

-0.02 

2... 

-    .16 

-   .06 

2.58 

(2.36) 

-    .02 

(+    -14) 

3... 

-   .04 

+   .01 

2.57 

2.54 

2.56 

2.50 

-    .01 

-    .04 

4... 

+   .11 

+   .02 

2.27 

2.40 

+   .29 

+   .10 

5... 

+   .09 

.00 

2.46 

2.55 

+   .10 

-    .05 

6... 

-   .10 

-    .06 

2.65 

2.49 

-   .09 

-1-   .03 

7... 

-   .24 

-    .02 

2.77 

2.51 

-    .21 

+  .01 

8... 

-   .13 

-   .03 

2.69 

2.53 

2.56 

2.52 

-   .13 

-   .01 

9... 

+   .13 

+   .02 

2.43 

2.58 

+   .13 

-   .06 

10... 

+   .22 

+   .03 

2.26 

2.51 

+   .30 

+   .01 

11... 

+   .18 

+   .02 

2.29 

2.49 

+   .30 

-    .05 

12... 

-    .28 

-    .14 

2.78 

2.36 

-    .19 

+   .08 

13... 

-   .22 

-   .03 

2.63 

2.38 

2.59 

2.44    • 

-    .04 

+   -06 

14... 

-   .15 

-   .04 

2.63 

2.44 

-    .04 

00 

15... 

-   .08 

-   .02 

2.63 

2.53 

-    .04 

-.  09 

16... 

-   .03 

.00 

2.54 

2.51 

+   .05 

-    .01 

17... 

-   .17 

-   .02 

2.61 

2.42 

-    .02 

+   .08 

18... 

-   .16 

-   .02 

2.68 

2.50 

2.59 

2.50 

-    .09 

.00 

19... 

-   .05 

.00 

2.57 

2.52 

+   .02 

-   .02 

20... 

.00 

-   .01 

2.55 

2.54 

+   .04 

-   .04 

21... 

+   .05 

.00 

2.55 

2.60 

-    .10 

-   .05 

22... 

+   .02 

-   .02 

2.54 

2.54 

-    .09 

+    .01 

23... 

.00 

-   .01 

2.56 

2.55 

2.45 

2.55 

-   .11 

.00 

24... 

+   .23 

+   .04 

2.20 

2.47 

+   .25 

+   .08 

25... 

+   -21 

.00 

2.39 

2.60    ' 

+   .06 

-    .05 

26... 

-    .01 

.00 

2.41 

2.40 

+   .12 

+   .04 

27... 

-   .20 

-   .03 

2.59 

2.36 

-   .06 

+  .08 

28... 

-   .08 

-    .02 

2.60 

2.50 

2.53 

2.44 

-   .07 

-   .06 

29... 

-   .09 

-   .02 

2.55 

2.44 

-    .02 

.00 

30... 

-   .03 

-    .01 

2.52 

2.48 

+   .01 

-   .04 

July     1... 

-   .14 

-    .03 

2.58 

2.41 

-    .24 

-   .08 

2... 

-   .10 

-   .01 

2.48 

2.37 

-    .14 

-   .04 

3... 

.00 

-   .05 

2.52 

(2-47) 

2.34 

2.33 

-    .18 

(-    -14) 

4... 

+   .19 

+   .04 

2.02 

2.25 

+   .32 

+   .08 

5... 

+   .17 

+   .01 

2.12 

2.30 

+   .22 

+   .03 

6... 

.00 

.00 

2.30 

2.30 

+   .11 

-    .06 

7... 

-   .19 

-   .01 

2.44 

2.24 

-   .03 

.00 

8... 

-   .14 

-   .01 

2.39 

2.24 

2.41 

2.24 

+   .02 

.00 

9... 

-   .21 

-   .04 

2.46 

2.21 

-    .05 

+   .03 

10... 

-    .21 

-    .04 

2.47 

2.22 

-    .06 

+   .02 

BAROMETRIC   PRESSURES    ON   THE   GREAT  LAKES  81 

TABLE  No.  19 — Continued. 


Date. 

Baro- 
metric 
cor- 
rection. 

Wind 
cor- 
rection. 

Observed 
eleva- 
tion 
570  +. 

Cor- 
rected 
eleva- 
tion 
570+. 

5-day 
observed 
mean 
570  +. 

5-day 
corrected 
mean 
570+. 

Residuals  from 
5-day  means. 

Obs. 

Cor. 

1910. 

feet. 

feet. 

feet. 

feet. 

feet. 

feet. 

feet. 

feet. 

July  11... 

-0.09 

-0.01 

2.42 

2.32 

+0.03 

+0.05 

12... 

-   .02 

-   .02 

2.42 

2.38 

+   .03 

-   .01 

13... 

-   .17 

-   .04 

2.62 

2.41 

2.45 

2.37 

-   .17 

-   .04 

14... 

+   .03 

.00 

2.38 

2.41 

+   .07 

-   .04 

15... 

-   .06 

-   .01 

2.39 

2.32 

+   .06 

+  .05 

16... 

-   .03 

+   -02 

2.18 

2.17 

-   .04 

+   .03 

17... 

+   .11 

+   .03 

1.95 

2.09 

+   .19 

+   .11 

18... 

+  .17 

+   .03 

2.03 

2.23 

2.14 

2.20 

+   .11 

-   .03 

19... 

+   .06 

.00 

2.23 

2.29 

-   .09 

-   .09 

20... 

-   .09 

.00 

2.29 

2.20 

-   .15 

.00 

21... 

-    .26 

-   .05 

2.39 

2.08 

+  .05 

+   .08 

22... 

-   .24 

-   .05 

2.47 

2.18 

-   .03 

-   .02 

23... 

-   .06 

-   .01 

2.28 

2.21 

2.44 

2.16 

+  .16 

-   .05 

24... 

-   .15 

-   .07 

2.42 

2.20 

+   .02 

-   .04 

25... 

-    .39 

-   .12 

2.64 

2.13 

-   .20 

+   .03 

26... 

-    .18 

-   .05 

2.46 

2.23 

-   .04 

+  .04 

27... 

-   .11 

-   .03 

2.49 

2.35 

-   .07 

-   .08 

28... 

-   .07 

-   .01 

2.35 

2.27 

2.42 

2.27 

+   .07 

.00 

29... 

-   .10 

-   .02 

2.38 

2.26 

+  .04 

+   .01 

30... 

-   .20 

-   .03 

2.45 

2.22 

-   .03 

+   .05 

31... 

-   .05 

.00 

2.36 

2.31 

+   .06 

-   .04 

Aug.     1.  .  . 

-   .06 

-   .04 

2.35 

2.25 

+   .13 

.00 

2... 

.00 

.00 

2.28 

2.28 

+   .20 

-   .03 

3... 

-   .03 

.00 

2.22 

2.19 

2.48 

2.25 

+   .26 

+   .06 

4... 

-   .30 

-   .12 

2.69 

2.27 

-   .21 

-   .02 

5... 

-   .31 

-   .26 

2.84 

2.27 

-   .36 

-   .02 

6... 

-    .21 

-   .06 

2.45 

2.18 

-   .20 

+   .01 

7... 

+  .06 

.00 

2.13 

2.19 

+   .12 

+   .00 

8... 

+   .10 

.00 

2.17 

2.27 

2.25 

2.19 

+   .08 

-   .08 

9..  . 

+   .01 

-   .01 

2.18 

2.18 

+   .07 

+   .01 

10... 

-   .14 

-   .03 

2.32 

2.15 

-   .07 

+   .04 

11... 

.00 

-   .01 

12... 

+   .05 

.00 

2!6i 

2!06 

+   .05 

!66 

13... 

-    .04 

.00 

2.08 

2.04 

2.06 

2.06 

-   .02 

+  .02 

14... 

-   .03 

.00 

2.10 

2.07 

-   .04 

-   .01 

15... 

+   .02 

.00 

2.04 

2.06 

+   .02 

.00 

16... 

+   -11 

.00 

1.98 

2.09 

+   -06 

-   .04 

17... 

+   .07 

-   .01 

2.01 

2.07 

+  .03 

-   .02 

18... 

-   .10 

-   .02 

2.16 

2.04 

2.04 

2.05 

-   .12 

+   .01 

19... 

+   .03 

-   .01 

2.05 

2.07 

-   .01 

-   .02 

20... 

.00 

-   .01 

2.01 

2.00 

+   .03 

+   .05 

82 


EFFECTS   OF   WINDS   AND   OF 
TABLE  No.  19 — Continued. 


Date. 

Baro- 
metric 
cor- 
rection. 

Wind 
cor- 
rection. 

Observe< 
eleva- 
tion 
570  +. 

Cor- 
rected 
eleva- 
tion 
570  +. 

5-day 
observec 
mean 
570  +. 

5-day 
correctec 
mean 
570+. 

Residuals  from 
5-day  means. 

Obs. 

Cor. 

1910. 

feet. 

feet. 

jtof. 

feet. 

** 

feet. 

feet. 

feet. 

Aug.  21.. 

-0.04 

0.00 

1.97 

1.93 

+0.16 

-0.04 

22.. 

-   .11 

-   .01 

2.11 

1.99 

+   .02 

-    .10 

23.. 

-   .17 

-   .03 

2.04 

1.84 

2.13 

1.89 

+  .09 

+   .05 

24.. 

-   .19 

-   .06 

2.10 

1.85 

+   .03 

+   -04 

25.. 

-   .45 

-   .14 

2.45 

1.86 

-   .32 

+   .03 

26.. 

-   .21 

-   .06 

2.30 

(2.03) 

-   .46 

(-   -09) 

27.. 

-   .03 

-   .01 

2.03 

1.99 

-   .19 

-    .05 

28.. 
29... 

+   -12 
+   .30 

+   .01 
+   .05 

1.81 
1.28 

1.94 
(1.63) 

1.84 

1.94 

+   .03 
+   .56 

.00 
(+   .31) 

30... 

+   -21 

.00 

1.73 

1.94 

+   .11 

.00 

31... 

+  .02 

-   .03 

1.88 

1.87 

-   .04 

+  .07 

Sept.    1... 

+  .09 

.00 

1.78 

1.87 

+   .12 

+   .07 

2... 

+   .16 

+  .02 

1.91 

(2.09) 

-   .01 

(-   -13) 

3... 

-   .13 

-   .05 

1.90 

1.94 

4... 

+   -05 

.00 

.... 

5... 

+   .02 

-    .02 

2!6i 

2^01 

-   .11 

-".Q7 

6... 

-   .25 

-   .13 

2.43 

(2.05) 

-   .35 

(-    .17) 

7... 

-   .17 

-   .03 

2.09 

1.89 

-   .01 

-    .01 

8... 

-   .08 

-   .01 

1.99 

1.90 

2.08 

1.88 

+  .09 

-   .02 

9... 

-   .20 

-   .01 

2.09 

1.88 

-   .01 

.00 

10... 

+   .04 

+   .01 

1.80 

1.85 

+   .28 

+   .03 

11... 

+   .03 

.00 

1.85 

1.88 

-   .05 

-    .06 

12... 

-   .04 

-   .02 

1.92 

1.86 

-    .12 

-   .04 

13... 

+   .04 

+   .01 

1.72 

1.77 

1.80 

1.82 

+   .08 

+   .05 

14... 

+   .06 

+   .01 

1.68 

1.75 

+   .12 

+   .07 

15... 

+   .03 

.00 

1.82 

1.85 

-    .02 

-   .03 

16... 

+   -05 

.00 

1.71 

1.76 

+   .02 

+   .01 

17... 

-   .07 

.00 

1.80 

1.73 

-   .07 

+   .04 

18... 

-   .03 

.00 

1.83 

1.80 

1.73 

1.77 

-   .10 

-   .03 

19... 

+   .13 

+   .02 

1.44 

(1.59) 

+   .29 

(  +   .18) 

20... 

-   .06 

-   .02 

1.87 

1.79 

-   .14 

-   .02 

21... 

+  -07 

.00 

1.72 

1.79 

-   .09 

-    .03 

22... 

+   .28 

+   .02 

1.27 

(1.57) 

+  .36 

(+   -19) 

23... 

+  .10 

.00 

1.66 

1.76 

1.63 

1.76 

-   .03 

.00 

24... 

+   .21 

+   .03 

1.52 

1.76 

+  .11 

.00 

25... 

-   .17 

-   .05 

1.96 

1.74 

-   .33 

+   .02 

26... 

+  .08 

+   .02 

1.52 

1.62 

+   .24 

.00 

27... 

-   .18 

-   .08 

2.09 

(1.83) 

-   .33 

(-   -21) 

28... 

-   .11 

-   .01 

1.82 

1.70 

1.76 

1.62 

-   .06 

-   .08 

29... 

+   .03 

.00 

1.57 

1.60 

+   .19 

+   .02 

30  .. 

-   .20 

-    .03 

1.80 

1.57 

-   .04 

+   .05 

BAROMETRIC   PRESSURES   ON   THE   GREAT   LAKES  83 

TABLE  No.  19— Continued. 


Date. 

Baro- 
metric 
cor- 
rection. 

Wind 
cor- 
rection. 

Observed 
eleva- 
tion 
570  +  . 

Cor- 
rected 
eleva- 
tion 
570  +. 

5-day 
observed 
mean 
570  +. 

5-day 
corrected 
mean 
570  +. 

Residuals  from 
5-day  means. 

Obs. 

Cor. 

1910. 

feet. 

feet. 

feet. 

feet. 

feet. 

feet. 

feet. 

feet. 

Oct.     1... 

-0.40 

-0.20 

2.42 

(1.82) 

-0.57 

(-0.24) 

2... 

+   .04 

.00 

.55 

1.59 

+  -30 

-   .01 

3... 

-   .01 

-   .02 

.62 

1.59 

1.85 

1.58 

+  .23 

-   .01 

4... 

-   .28 

-   .10 

.94 

1.56 

-   .09 

+   .02 

5... 

-   .26 

-   .03 

.74 

(1.45) 

+  .11 

(+   .13) 

6... 

-   .13 

.00 

.86 

1.73 

+   .01 

+  -03 

7... 

+   .09 

+   .01 

.66 

1.76 

+  .21 

.00 

8... 

-   .06 

-   .01 

.83 

1.76 

1.87 

1.76 

+   .04 

.00 

9... 

-   .29 

-   .02 

2.08 

1.77 

-   .21 

-   .01 

10... 

-   .08 

-   .06 

1.92 

1.78 

-   .05 

-   .02 

11... 

-   .25 

-   .16 

2.31 

(1.90) 

-   .46 

(-   .14) 

12... 

+   .16 

+   .02 

1.53 

1.71 

+   .32 

+  .05 

13... 

+   .21 

.00 

1.50 

1.71 

1.85 

1.76 

+   .35 

+   .05 

14... 

-   .10 

-   .02 

1.93 

1.81 

-   .08 

-  .05 

15... 

-   .14 

-   .05 

1.99 

1.80 

-   .14 

-   .04 

16... 

-   .23 

-   .04 

1.97 

1.70 

-   .24 

.00 

17... 

+   .09 

.00 

1.65 

1.74 

+   .08 

-   .04 

18... 

-   .06 

-   .01 

1.76 

1.69 

1.73 

1.70 

-   .03 

+   .01 

19... 

-   .07 

-   .02 

1.78 

1.69 

-   .05 

+   .01 

20... 

+   .18 

+   .03 

1.47 

1.68 

+   .26 

+  .02 

21... 

+   .16 

+   .02 

1.36 

(1.54) 

+   .72 

(+   .19) 

22... 

-   .48 

-   .22 

3.11 

(2.41) 

-1.03 

(-   .68) 

23... 

-   .26 

-   .04 

2.04 

1.74 

2.08 

1.73 

+   .04 

-   .01 

24... 

-    .21 

-   .08 

2.05 

1.76 

+   .03 

-   .03 

25... 

-   .14 

-   .03 

1.85 

1.68 

+  .23 

+  .05 

26... 

-   .09 

.00 

1.69 

1.60 

+   .37 

+  .07 

27... 

-    .35 

-   .22 

2.63 

(2.06) 

-   .57 

(-   -39) 

28... 

29..  . 

-   .21 
-   .08 

-   .04 
.00 

l!82 

i.74 

2.06 

1.67 

+  ".24: 

-   .07 

30... 

-   .10 

-   .11 

2.34 

(2.13) 

-   .28 

(-   -46) 

31... 

-    .23 

-   .06 

1.80 

(1.51) 

+   .26 

(+   -16) 

84 


EFFECTS   OF  WINDS  AND   OF 


TABLE  No.  20 — Observed  and  corrected  elevations  of  water  surface  at  the  Cleveland  Gage  on 

Lake  Erie. 


Date. 

Baro- 
metric 
cor- 
rection. 

Wind 
cor- 
rection. 

Observed 
eleva- 
tion 
570  +. 

Cor- 
rected 
eleva- 
tion. 
570+.^ 

5-day 
observed 
mean 
570+. 

5-day 
correctec 
mean 
570+  . 

Residuals  from 
5-day  means. 

Obs. 

Cor. 

1910. 

feet. 

feet. 

feet. 

feet. 

feet. 

feet. 

/•*. 

feet. 

June    1.  .  . 

-0.08 

-0.01 

2.57 

2.64 

+0.08 

0.00 

2... 

+   .03 

.00 

2.60 

2.63 

+   .05 

+  .01 

3... 

-   .07 

-   .01 

2.71 

2.63 

2.65 

2.64 

-   .06 

+   .01 

4... 

-   .01 

.00 

2.70 

2.69 

-   .05 

-   .05 

5... 

-   .06 

.00 

2.67 

2.61 

-   .02 

+   .03 

6... 

-   .02 

.00 

2.62 

2.60 

+   .01 

+   .02 

7... 

+   .02 

-   .02 

2.59 

2.59 

+   .04 

+   .03 

8... 

+  .05 

.00 

2.58 

2.63 

2.63 

2.62 

+   .05 

-   .01 

9... 

-   .02 

+   .01 

2.66 

2.65 

-   .03 

-   .03 

10... 

-   .09 

+  .02 

2.72 

2.65 

-   .09 

-    .03 

11... 

-   .11 

+  .02 

2.75 

2.66 

-    .11 

+   .01 

12... 

+   .10 

-   .01 

2.59 

2.68 

+  .05 

-   .01 

13... 

+   .09 

.00 

2.63 

2.72 

2.64 

2.67 

+   .01 

-   .05 

14... 

+   .04 

.00 

2.62 

2.66 

+   .02 

+   .01 

15... 

+   .01 

.00 

2.63 

2.64 

+   .01 

+  .03 

16... 

-   .01 

.00 

2.64 

2.63 

-   .01 

.00 

17... 

-f   .04 

-   .01 

2.62 

2.65 

+   .01 

-   .02 

18... 

+   .02 

-   .02 

2.61 

2.61 

2.63 

2.63 

+   .02 

+   .02 

19... 

.00 

.00 

2.65 

2.65 

-   .02 

-   .02 

20... 

-   .02 

.00 

2.62 

2.60 

+   .01 

+   .03 

21... 

-   .03 

.00 

2.61 

2.58 

-   .02 

-   .01 

22... 

-   .02 

.00 

2.59 

2.57 

.00 

.00 

23... 

-   .02 

.00 

2.58 

2.56 

2.59 

2.57 

+  .01 

+   .01 

24... 

-   .09 

.00 

.... 

.... 

.... 

.... 

25... 

-   .04 

+   .02 

26... 

+  .05 

.00 

27... 

+   .09 

.00 

.... 

.... 

28... 

.00 

.00 

2.43 

2.43 

29... 

+   .01 

.00 

2.42 

2AZ 

+"!6i 

.66 

30... 

-   .01 

.00 

2.44 

2.43 

-   .01 

.00 

July     1... 

+  .02 

.00 

2.38 

2.40 

+   .05 

+   .01 

2... 

+   .04 

.00 

2.40 

2.44 

+  .03 

-   .03 

3... 

-   .01 

.00 

2.40 

2.39 

2.43 

2.41 

+  .03 

+   .02 

4... 

-   .14 

+  .01 

2.56 

2.43 

-   .13 

-   .02 

6... 

-   .02 

+  .01 

2.41 

2.40 

+   .02 

+   .01 

6... 

+  .06 

.00 

2.36 

2.42 

-   .02 

+   .01 

7... 

+  .07 

.00 

2.36 

2.43 

-   .02 

.00 

8... 

+   ,09 

.00 

2.37 

2.46 

2.34 

2.43 

-   .03 

-   .03 

9... 

+  .13 

.00 

2.31 

2.44 

+  .03 

-   .01 

10... 

+  .10 

-   .01 

2.31 

2.40 

+  .03 

+   .03 

BAROMETRIC   PRESSURES   ON   THE   GREAT  LAKES 
TABLE  No.  20— Continued. 


85 


Date. 

Baro- 
metric 
cor- 
rection. 

Wind 
cor- 
rection. 

Observed 
eleva- 
tion 
570  +. 

Cor- 
rected 
eleva- 
tion 
570  +. 

5-day 
sbserved 
mean 
570  +. 

5-day 
corrected 
mean 
570+  . 

Residuals  from 
5-day  means. 

Obs. 

Cor. 

1910. 

feet. 

**. 

feet. 

feet. 

feet. 

feet. 

feet. 

feet. 

July  11... 

+0.07 

0.00 

12... 

+   .06 

+   .01 

2.30 

2.37 

+0.12 

+6!  08 

13... 

+   .03 

.00 

2.38 

2.41 

2.42 

2.45 

+   .04 

+   .04 

14... 

.00 

.00 

2.49 

2.49 

-   .07 

-   .04 

15... 

+  .02 

.00 

2.51 

2.53 

-   .09 

-   .08 

16... 

-    .01 

.00 

2.58 

(2.57) 

-   .03 

(-   .17) 

17... 

-   .08 

+   .01 

2.64 

(2.57) 

-   .09 

(-  -17) 

18... 

-   .11 

-   .01 

2.54 

2.42 

2.55 

2.40 

+   .01 

-   .02 

19... 

-   .06 

.00 

2.43 

2.37 

+   .12 

+  .03 

20... 

+  .05 

.00 

.... 

.... 

21... 

+  .13 

.00 

2.28 

2.41 

+   .05 

-   .02 

22... 

+   .10 

.00 

2.29 

2.39 

+  .04 

.00 

23... 

+   .05 

.00 

2.33 

2.38 

2.33 

2.39 

.00 

+   .01 

24... 

+   .12 

+   .03 

.... 

.... 

25... 

+   .15 

-   .02 

2A1 

(2.54) 

-  !os 

(-   .15) 

26... 

+   .06 

.00 

2.43 

2.49 

-   .03 

-   .04 

27... 

+   .06 

.00 

2.33 

2.39 

+  .07 

+  .06 

28... 
29... 

.00 

+   .06 

.00 
.00 

2.42 
2.37 

2.42 
2.43 

2.40 

2.45 

-   .02 
+  .03 

+  .03 
+   .02 

30... 

+   .03 

.00 

2.47 

2.50 

-   .07 

-   .05 

31... 

-   .05 

.00 

2.40 

(2.35) 

.00 

(+   .10) 

Aug.    1... 

+   .05 

.00 

2.37 

2.42 

-   .02 

-   .01 

2... 

+   .02 

.00 

2.40 

2.42 

-   .05 

-   .01 

3... 

+   .06 

.00 

2.37 

2.43 

2.35 

2.41 

-   .02 

-   .02 

4... 

+   .10 

-   .02 

2.35 

2.43 

.00 

-   .02 

5... 

+   .10 

-   .04 

2.27 

2.33 

+   .08 

+  .08 

6... 

+   .09 

-   .01 

2.28 

2.36 

-   .03 

-   .07 

7... 

+   .01 

.00 

2.28 

2.29 

-   .03 

.00 

8... 

-   .01 

.00 

2.23 

2.22 

2.25 

2.29 

+  .02 

+   .07 

9... 

+   .09 

.00 

2.18 

2.27 

+  .07 

+   .02 

10... 

+  .07 

.00 

2.26 

2.33 

-   .01 

-    .04 

11... 

.00 

.00 

2.26 

2.26 

.00 

+  .02 

12... 

+   .02 

-   .01 

2.29 

2.30 

-   .03 

-   .02 

13... 

+   .09 

.00 

2.26 

2.28 

14... 

+   .08 

.00 

.... 

15... 

+  .05 

.00 

2^24 

2.29 

+  ".02 

-   .01 

16... 

+  .03 

.00 

2.25 

2.28 

-   .01 

+  .02 

17... 

+   .08 

+   .01 

2.21 

2.30 

+  .03 

.00 

18... 

+  .09 

.00 

2.24 

2.33 

2.24 

2.30 

.00 

-   .03 

19... 

-   .01 

-   .01 

2.29 

2.27 

-   .05 

+   .03 

20... 

+   -07 

.00 

2.23 

2.30 

+  .01 

.00 

86 


EFFECTS   OF   WINDS   AND   OF 
TABLE  No.  20 — Continued. 


Date. 

Baro- 
metric 
cor- 
rection. 

Wind 
cor- 
rection. 

Observed 
eleva- 
tion 
570  +. 

Cor- 
rected 
eleva- 
tion 
570  +. 

5-day 
observed 
mean 
570  +. 

5-day 
corrected 
mean 
570  -K 

Residuals  from 
5-day  means. 

Obs. 

Cor. 

1910. 

feet. 

feet. 

feet. 

feet. 

feet. 

feet. 

feet. 

feet. 

Aug.  21... 

+0.12 

0.00 

2.17 

2.29 

-0.09 

-0.03 

22... 

+  .16 

+  .01 

2.08 

2.25 

.00 

+  .01 

23... 

+   .16 

+  .01 

2.09 

2.26 

2.08 

2.26 

-   .01 

.00 

24... 

+   .20 

+  .02 

2.03 

2.25 

-I-   .05 

+  .01 

25... 

+  .24 

.00 

2.02 

2.26 

+  .06 

.00 

26... 

+  .05 

-   .02 

2.12 

2.15 

.00 

-   .03 

27... 

+  .02 

.00 

2.04 

2.06 

+   .08 

+   .06 

28... 
29... 

-   .03 
-   .11 

.00 
+   .03 

2.11 
2.26 

2.08 
2.18 

2.12 

2.12 

+   .01 
-.   14 

+   .04 
-   .06 

30... 

+   .03 

+  .02 

2.06 

2.11 

+  .06 

+   .01 

31... 

.00 

.00 

2.11 

2.11 

+   .01 

+   .01 

Sept.    1... 

-   .11 

+  .01 

2.19 

2.09 

-   .06 

+   .05 

2... 

-   .03 

.00 

2.14 

2.11 

-   .01 

-1-   .03 

3... 

+  .09 

.00 

2.05 

2.14 

2.13 

2.14 

+   .08 

.00 

4... 

-   .01 

.00 

2.20 

2.19 

-   .07 

-   .05 

5... 

+  .06 

+  .01 

2.08 

2.15 

+   .05 

-   .01 

6... 

+  .12 

-   .02 

2.03 

2.13 

+  .08 

+  .01 

7... 

+   .02 

.00 

2.10 

2.12 

+  .01 

+  .02 

8... 

+  .08 

.00 

2.12 

2.20 

2.11 

2.14 

-   .01 

-   .06 

9... 

-  .03 

-   .02 

2.21 

2.16 

-   .10 

-   .02 

10... 

-  .03 

.00 

2.10 

2.07 

+  .01 

+  .07 

11... 

+   .01 

.00 

2.07 

2.08 

-   .01 

-   .07 

12... 

4-  .03 

.00 

2.02 

2.05 

+  .04 

-   .04 

13... 

-   .12 

-   .01 

2.13 

2.00 

2.06 

2.01 

-  .07 

+  .01 

14... 

-   .12 

-   .01 

2.11 

1.98 

-  .05 

-f   .03 

15... 

-   .06 

.00 

1.98 

1.92 

+   .08 

+  .09 

16... 

-   .05 

.00 

17... 

+   .04 

.00 

18... 

.00 

.00 

1.93 

1.93 

1.97 

1.96 

+  .04 

+  .03 

19... 

-   .05 

.00 

2.06 

2.01 

-   .09 

-   .05 

20... 

+   .02 

.00 

1.93 

1.95 

+   .04 

+  .01 

21... 

-   .06 

-  .01 

1.99 

1.92 

-   .04 

+  .01 

22... 

-   .07 

.00 

2.05 

1.98 

-   .10 

-  .05 

23... 

+   .02 

.00 

1.91 

1.93 

1.95 

1.93 

+   .04 

.00 

24... 

.00 

+   .01 

1.90 

1.91 

+   .05 

+   .02 

25... 

+  .02 

.00 

1.89 

1.91 

+  .06 

+   .02 

26... 

-  .05 

.00 

27... 

+  .06 

.00 

iise 

i!<J2 

!oo 

-   !02 

28  .. 

.00 

.00 

1.89 

1.89 

1.86 

1.90 

-   .03 

+   .01 

29... 

-   .01 

.00 

1.90 

1.89 

-   .04 

+   .01 

30... 

+  .13 

.00 

1.79 

1.92 

+  .07 

-   .02 

BAROMETRIC   PRESSURES   ON   THE    GREAT  LAKES 
TABLE  No.  20 — Continued. 


87 


Date. 

Baro- 
metric 
cor- 
rection. 

Wind 
cor- 
rection. 

Observed 
eleva- 
tion 
570  +. 

Cor- 
rected 
eleva- 
tion 
570+  . 

5-day 
>bserved 
mean 
570  +. 

5-day 
corrected 
mean 
570  +. 

Residuals  from 
5-day  means. 

Obs. 

Cor. 

1910. 

feet. 

feet. 

feet. 

feet. 

feet. 

feet. 

feet. 

feet. 

Oct.     1... 

+0.13 

-0.05 

1.79 

1.87 

-0.03 

+0.02 

2... 

+   .01 

.00 

1.86 

1.87 

-   .10 

+   .02 

3... 

+   .13 

+   .02 

1.74 

1.89 

1.76 

1.89 

+   .02 

.00 

4... 

+   .21 

+   .03 

1.64 

1.88 

+   .12 

+   .01 

5... 

+  .16 

+   .02 

1.78 

1.96 

-   .02 

-  .07 

6... 

-   .03 

-   .04 

2.28 

(2.21) 

-   .18 

(-   -20) 

7... 

-   .10 

-   .01 

2.13 

2.02 

-   .03 

-   .01 

8... 

.00 

.00 

2.03 

2.03 

2.10 

2.01 

+  .07 

-   .02 

9... 

-   .01 

-   .02 

2.07 

2.04 

+   .03 

-   .03 

10... 

-   .03 

.00 

1.97 

1.94 

+   .13 

+  .07 

11... 

+  .05 

-    .01 

1.79 

1.83 

+   .12 

+  .05 

12... 

-   .13 

+   .01 

2.07 

1.95 

-   .16 

-   .07 

13... 

-   .08 

+   .01 

1.91 

1.88 

14... 

-   .02 

.00 

iiss 

l!86 

+   .03 

+  !62 

15... 

-   .02 

.00 

1.90 

1.88 

+  .01 

.00 

16... 

-    .02 

-   .01 

1.89 

1.86 

+   .07 

+  .05 

17..  . 

-   .07 

.00 

1.98 

1.91 

-   .02 

.00 

18... 

+   .02 

.00 

1.94 

1.96 

1.96 

1.91 

+   .02 

-   .05 

19... 

+   .03 

+   .01 

.... 

.... 

20... 

-   .15 

.00 

2^05 

1.90 

-"69 

+  .01 

21... 

-   .08 

.00 

2.09 

2.01 

-   .26 

-   .06 

22... 

+  .14 

-   .04 

1.56 

(1.66) 

+  .27 

(+   -29) 

23... 

+   .05 

-    .02 

1.92 

1.95 

1.83 

1.95 

-   .09 

.00 

24... 

+  .08 

+  .02 

1.64 

(1.74) 

+  .19 

(+   -21) 

25... 

-   .03 

-    .02 

1.93 

1.88 

-   .10 

+  .07 

26... 

+   .04 

.00 

1.85 

1.89 

-   .13 

-   .03 

27... 

+   .12 

-   .06 

1.74 

1.80 

-    .02 

+   .06 

28... 
29... 

+   .07 
+   .02 

-   .06 
-   .04 

1.85 
1.94 

1.86 
1.92 

1.72 

1.86 

-   .13 
-   .22 

.00 
-   .06 

30... 

+   .20 

+   .03 

1.33 

(1.56) 

+   .39 

(+   .30) 

31... 

+   .17 

.00 

1.64 

1.81 

+   .08 

+   .05 

88 


EFFECTS   OF   WINDS   AND    OF 


TABLE  No.  21 — Observed  and  corrected  elevations  of  water  surface  at  the  Milwaukee  Gage 
Lake  Michigan-Huron. 


Date. 

Baro- 
metric 
cor- 
rection. 

Wind 
cor- 
rection. 

Observed 
eleva- 
tion 
579+. 

Cor- 
rected 
eleva- 
tion 
579+. 

5-day 
observec 
mean 
579+. 

5-day 
correctec 
mean 
579+. 

Residuals  from 
5-day  means. 

Obs. 

Cor. 

1911. 

feet. 

feet. 

feet. 

feet. 

feet. 

feet. 

feet. 

feet. 

June    1.  .  . 

+0.20 

0.00 

0.79 

0.99 

+0.21 

-0.01 

2... 

+  .03 

.00 

.96 

.99 

+  .04 

-   .01 

3... 

-   .11 

.00 

1.06 

.95 

1.00 

.98 

-   .06 

+   .03 

4... 

-   .16 

.00 

1.13 

.97 

-   .13 

+   .01 

5... 

-   .06 

.00 

1.04 

.98 

-   .04 

.00 

6... 

-   .12 

.00 

1.11 

.99 

-   .04 

+   .05 

7... 

-   .06 

.00 

1.03 

.97 

+  .04 

+   .07 

8... 

-    .03 

.00 

1.10 

1.07 

1.07 

1.04 

-   .03 

-   .03 

9... 

-   .03 

.00 

1.12 

1.09 

-   .05 

-   .05 

10... 

+   .05 

.00 

1.01 

1.06 

+   .06 

-   .02 

11... 

+   .06 

.00 

.96 

1.02 

+   .01 

+   .05 

12... 

+   .06 

.00 

.94 

1.00 

+   .03 

+   .07 

13... 

+   .14 

.00 

.97 

1.11 

.97 

1.07 

.00 

-   .04 

14... 

+   .13 

.00 

.98 

1.11 

-    .01 

-   .04 

15... 

+   .09 

.00 

1.02 

1.11 

-   .05 

-   .04 

16... 

+  .04 

.00 

.07 

1.11 

-    .01 

-   .04 

17... 

-   .07 

.00 

.11 

1.04 

-   .05 

+   .03 

18... 

+   .02 

.00 

.04 

1.06 

1.06 

1.07 

+   .02 

+   .01 

19... 

+   .07 

.00 

.00 

1.07 

+   .06 

.00 

20... 

+  .03 

.00 

.06 

1.09 

.00 

-   .02 

21... 

+  .03 

.00 

1.05 

1.08 

+   .09 

+   .01 

22... 

+   .06 

.00 

1.00 

1.06 

+   .14 

+   .03 

23... 

-   .08 

.00 

1.16 

1.08 

1.14 

1.09 

-   .02 

+   .01 

24... 

-   .17 

.00 

1.27 

1.10 

-   .13 

-   .01 

25... 

-   .12 

.00 

1.23 

1.11 

-    .09 

-   .02 

26... 

-   .01 

.00 

1.13 

1.12 

-   .06 

-   .02 

27... 

+   .10 

.00 

.99 

1.09 

+   .08 

+   .01 

28... 

+   .07 

.00 

.99 

1.06 

1.07 

1.10 

+   .08 

+   .04 

29... 

-   .04 

.00 

1.13 

1.09 

-   .06 

+   .01 

30... 

+  .02 

.00 

1.13 

1.15 

-   .06 

-   .05 

July     1... 

+   .08 

.00 

1.00 

1.08 

+   .01 

-   .01 

2... 

+   .15 

.00 

.93 

1.08 

+   .08 

-   .01 

3... 

+   .06 

.00 

1.00 

1.06 

1.01 

1.07 

+   .01 

+   .01 

4... 

-   .02 

.00 

1.07 

1.05 

-   .06 

+   .02 

5... 

+  .01 

.00 

1.06 

1.07 

-   .05 

.00 

6... 

+   .05 

.00 

1.05 

1.10 

+   .01 

+   .03 

7... 

-   .01 

.00 

1.15 

1.14 

-   .09 

-   .01 

8... 

+   .01 

.00 

1.16 

1.17 

1.06 

1.13 

-   .10 

-   .04 

9... 

+   .10 

.00 

1.00 

1.10 

+   .06 

+   .03 

10... 

+   .18 

.00 

.94 

(1.12) 

+   .12 

(+   -01) 

BAROMETRIC   PRESSURES   ON   THE   GREAT  LAKES  89 

TABLE  No.  21— Continued. 


Date. 

Baro- 
metric 
cor- 
rection. 

Wind 
cor- 
rection. 

Observed 
eleva- 
tion 
579+. 

Cor- 
rected 
eleva- 
tion 
579+. 

5-day 
observed 
mean 
579+. 

5-day 
correctec 
mean 
579+. 

Residuals  from 
5-day  means. 

Obs. 

Cor. 

1911. 

** 

** 

/«*. 

feet. 

feet. 

feet. 

** 

feet. 

July  11... 

+0.16 

0.00 

0.85 

1.01 

+0.02 

-0.01 

12... 

+   .14 

.00 

.81 

.95 

+   .06 

+   .05 

13... 

+  .13 

.00 

.87 

1.00 

.87 

1.00 

.00 

.00 

14... 

+   .10 

.00 

.89 

.99 

-   .02 

+   .01 

15... 

+  -12 

.00 

.92 

1.04 

-   .05 

-   .04 

16... 

+   .04 

.00 

.95 

.99 

-   .04 

+  .01 

17... 

+   .13 

.00 

.88 

1.01 

+  .03 

-   .01 

18... 

+  .15 

.00 

.89 

1.04 

.91 

1.00 

+   .02 

-   .04 

19... 

+   .05 

.00 

.97 

1.02 

-   .06 

-   .02 

20... 

+   .12 

.00 

.84. 

.96 

+  .07 

+  .04 

21..  . 

+   .12 

.00 

.76 

.88 

-   .06 

-   .02 

22... 

+   .16 

.00 

.77 

.93 

-   .07 

-  .07 

23... 

+  .02 

.00 

.82 

.84 

.70 

.86 

-   .12 

+  .02 

24... 

+   .21 

.00 

.65 

.86 

+  .05 

.00 

25... 

+  .30 

.00 

.48 

.78 

+  .22 

+  .08 

26... 

+   .12 

.00 

.74 

.86 

+   .06 

-   .01 

27... 

+  .07 

.00 

.81 

.88 

-   .01 

-   .03 

28... 

+   .03 

.00 

.84 

.87 

-   .04 

-   .02 

29... 

+   .06 

.00 

.80 

.86 

.80 

.85 

.00 

-   .01 

30... 

+   .04 

.00 

.78 

.82 

+   .02 

+   .03 

31... 

-   .02 

.00 

.83 

.81 

-   .03 

+  .04 

Aug.    1... 

-   .03 

.00 

.91 

.88 

-   .03 

-   .02 

2... 

+   .06 

.00 

.83 

.89 

+   .05 

-   .03 

3... 

-   .06 

.00 

.86 

.80 

.88 

.86 

+  .02 

+   .06 

4... 

-   .06 

.00 

.93 

.87 

—   .05 

-   .01 

5... 

.00 

.00 

.85 

.85 

+   .03 

+  .01 

6... 

-    .04 

.00 

.87 

.83 

+   .02 

+  .06 

7... 

-   .07 

.00 

.95 

.88 

-   .06 

+   .01 

8... 

+   .12 

.00 

.80 

.92 

.89 

.89 

+  .09 

-   .03 

9... 

+   .10 

.00 

.77 

.87 

+  -12 

+  .02 

10... 

-   .09 

.00 

1.06 

.97 

-   .17 

-   .08 

11... 

-   .05 

.00 

1.06 

1.01 

-   .10 

-   .03 

12... 

-   .05 

.00 

.95 

(.90) 

+  .01 

(+  .08) 

13... 

+   .01 

.00 

.96 

.97 

.96 

.98 

.00 

+  .01 

14... 

+   .04 

.00 

.95 

.99 

+  .01 

-   .01 

15... 

+  .07 

.00 

.86 

.93 

+  .10 

+  .05 

16... 

+   .12 

.00 

.84 

.96 

+  .01 

-   .03 

17... 

+   .10 

.00 

.88 

.98 

-   .03 

-   .05 

18... 

+  .08 

.00 

.83 

.91 

.85 

.93 

+  .02 

+   .02 

19... 

+   .06 

.00 

.86 

.92 

-  .01 

+  .01 

20... 

+   .04 

.00 

.85 

.89 

.00 

+  .04 

1 

90 


EFFECTS   OF  WINDS  AND    OF 

TABLE  No.  21— Continued. 


Date. 

Baro- 
metric 
cor- 
rection. 

Wind 
cor- 
rection. 

Observed 
eleva- 
tion 
579+. 

Cor- 
rected 
eleva- 
tion 
579+. 

5-day 
observed 
mean 
579+. 

5-day 
correctec 
mean 
579+. 

Residuals  from 
5-day  means. 

Obs. 

Cor. 

1911. 

feet. 

feet. 

feet. 

feet. 

feet. 

feet. 

feet. 

feet. 

Aug.  21... 

+0.08 

0.00 

0.73 

0.81 

+0.06 

+0.01 

22... 

+  .01 

.00 

.84 

.85 

-   .05 

-   .03 

23... 

+  .04 

.00 

.77 

.81 

.79 

.82 

+   .02 

+   .01 

24... 

-   .03 

.00 

.83 

.80 

-   .04 

+   .02 

25... 

+  .02 

.00 

.80 

.82 

-   .01 

.00 

26... 

+  .03 

.00 

.76 

.79 

-   .02 

-   .05 

27... 

-   .02 

.00 

.77 

.75 

-   .03 

-   .01 

28... 
29... 

+   .08 
+   .04 

.00 
.00 

.67 
.70 

.75 

.74 

.74 

.74 

+  .07 
+  .04 

-   .01 
.00 

30... 

-   .09 

.00 

.75 

.66 

-   .01 

+   .08 

31... 

-   .06 

.00 

.79 

.73 

-   .05 

+  .01 

Sept.    1... 

+  .06 

.00 

.67 

.73 

+   .05 

+  .01 

2... 

+  .10 

.00 

.66 

.76 

+  .06 

-   .02 

3... 

+   .06 

.00 

.67 

.73 

.72 

.74 

+  .05 

+   .01 

4... 

-   .04 

.00 

.78 

.74 

-   .06 

.00 

5... 

-  .08 

.00 

.80 

.72 

-   .08 

+   .02 

6... 

-   .08 

.00 

.92 

.84 

-   .01 

+   .01 

7... 

-   .21 

.00 

1.08 

.87 

-   .17 

-   .02 

8... 

-   .14 

.00 

.99 

.85 

.91 

.85 

-   .08 

.00 

9... 

+  .02 

.00 

.82 

.84 

+   .09 

+  .01 

10... 

+  .10 

.00 

.74 

.84 

+  .17 

+  .01 

11... 

+   .08 

.00 

.71 

.79 

-   .01 

-  .06 

12... 

+   .06 

.00 

.73 

.79 

-   .03 

-   .06 

13... 

-   .04 

.00 

.76 

.72 

.70 

.73 

-  .06 

+   .01 

14... 

-   .07 

.00 

.70 

.63 

.00 

+   .10 

15... 

+   .11 

.00 

.61 

.72 

+  .09 

+   .01 

16... 

+  .11 

.00 

.59 

.70 

+  .11 

+  .02 

17... 

-   .08 

.00 

.83 

.75 

-   .13 

-   .03 

18... 

-   .02 

.00 

.79 

.77 

.70 

.72 

-   .09 

-   .05 

19... 

+   .10 

.00 

.60 

.70 

+   .10 

+   .02 

20... 

.00 

.00 

.70 

.70 

.00 

+   .02 

21... 

-   .09 

.00 

.78 

.69 

-   .04 

+   .03 

22... 

+   .08 

.00 

.64 

.72 

+  .10 

.00 

23... 

+   .06 

.00 

.63 

.69 

.74 

.72 

+  .11 

+  .03 

24... 

-   .05 

.00 

.82 

.77 

-   .08 

-   .05 

25... 

-   .06 

.00 

.81 

.75 

-   .07 

-   .03 

26... 

.00 

.00 

.65 

.65 

+  .07 

+  .03 

27... 

+   .02 

.00 

.67 

.69 

+   .05 

-   .01 

28... 

-   .07 

.00 

.75 

.68 

.72 

.68 

-   .03 

.00 

29... 

-   .06 

.00 

.86 

(.80) 

-   .14 

(-   .12) 

30... 

.00 

.00 

.68 

.68 

+  .04 

.00 

BAROMETRIC   PRESSURES   ON   THE   GREAT  LAKES 


91 


TABLE  No.  22 — Observed  and  corrected  elevations  of  water  surface  at  the  Harbor-Beach  Gage 

on  Lake  Huron. 


Date. 

Baro- 
metric 
cor- 
rection. 

Wind 
cor- 
rection. 

Observed 
eleva- 
tion 
579+. 

Cor- 
rected 
eleva- 
tion 
579+. 

5-day 
observed 
mean 
579+. 

5-day 
correctec 
mean 
579+. 

Residuals  from 
5-day  means. 

Obs. 

Cor. 

1911. 

feet. 

feet. 

feet. 

feet. 

feet. 

feet. 

feet. 

feet. 

June    1..  . 

-0.11 

0.00 

1.02 

0.91 

-0.11 

+0.01 

2... 

-   .03 

.00 

.90 

.87 

+   .01 

+  .05 

3... 

+   .05 

.00 

.86 

.91 

0.91 

0.92 

+  .05 

+   .01 

4... 

+   .09 

.00 

.86 

.95 

+  .05 

-   .03 

5... 

+   .04 

.00 

.93 

.97 

-   .02 

-   .05 

6... 

+   .05 

.00 

.94 

.99 

-   .02 

-   .02 

7... 

.00 

.00 

.94 

.94 

-   .02 

+  .03 

8... 

+   .02 

.00 

.95 

.97 

.92 

.97 

-   .03 

.00 

9... 

+   .13 

.00 

.86 

.99 

+   .06 

-   .02 

10... 

+   .07 

.00 

.91 

.98 

+   .01 

-   .01 

11... 

+   .04 

.00 

.97 

1.01 

+   .09 

.00 

12... 

.00 

.00 

1.06 

1.06 

.00 

-   .05 

13... 

-  .11 

.00 

1.10 

.99 

1.06 

1.01 

-   .04 

+   .02 

14... 

-  .11 

.00 

1.11 

1.00 

-   .05 

+  .01 

15... 

-   .09 

.00 

1.07 

.98 

-   .01 

+   .03 

16... 

-   .05 

.00 

17... 

-   .01 

.00 

18... 

-   .06 

.00 

!6e 

i!66 

1.07 

1.00 

+"!6i 

.66 

19... 

-   .04 

.00 

.06 

1.02 

+   .01 

-   .02 

20... 

-    .09 

.00 

.08 

.99 

-   .01 

+   .01 

21... 

-   .02 

.00 

.03 

1.01 

-   .07 

+   .01 

22... 

-   .02 

.00 

.05 

1.03 

-   .09 

-   .01 

23... 

.00 

.00 

.04 

1.04 

.96 

1.02 

-   .08 

-   .02 

24... 

+   .10 

.00 

.82 

(.92) 

+   .14 

(+   -10) 

25... 

+   .10 

.00 

.88 

.98 

+   .08 

+  .04 

26... 

+   .07 

.00 

.89 

.96 

+   .08 

+  .01 

27... 

-   .04 

.00 

.98 

.94 

-   .01 

+   .03 

28... 

-   .12 

.00 

1.14 

1.02 

.97 

.97 

-   .17 

-   .05 

29... 

.00 

.00 

.98 

.98 

-   .01 

-   .01 

30... 

+  .08 

.00 

.86 

.94 

+   .11 

+   .03 

July     1... 

+   .04 

.00 

.95 

.99 

+   .02 

.00 

2... 

-   .01 

.00 

1.00 

.99 

-   .03 

.00 

3... 

-   .05 

.00 

1.02 

.97 

.97 

.99 

-   .05 

+   .02 

4... 

+   .02 

.00 

.98 

1.00 

-   .01 

-   .01 

5... 

+   .09 

.00 

.91 

1.00 

+   .06 

-   .01 

6... 

-   .07 

.00 

1.06 

.99 

-   .12 

-   .01 

7... 

-   .01 

.00 

1.00 

.99 

-   .06 

-   .01 

8... 

+  .10 

.00 

.86 

.96 

.94 

.98 

+   .08 

+   .02 

9... 

+   .13 

.00 

.85 

.98 

+   .09 

.00 

10... 

+   .03 

.00 

.94 

.97 

.00 

+   .01 

92 


EFFECTS   OF   WINDS   AND   OF 
TABLE  No.  22— Continued. 


Date. 

Baro- 
metric 
cor- 
rection. 

Wind 
cor- 
rection. 

Observed 
eleva- 
tion 
5794. 

Cor- 
rected 
eleva- 
tion 
5794. 

5-day 
observed 
mean 
5794. 

5-day 
correctec 
mean 
5794. 

Residuals  from 
5-day  means. 

Obs. 

Cor. 

1911. 

feet. 

feet. 

feet. 

feet. 

feet. 

feet. 

feet. 

feet. 

July  11... 

+0.02 

0.00 

1.01 

1.03 

-0.02 

-0.04 

12... 

-   .06 

.00 

1.08 

1.02 

-   .09 

-   .03 

13... 

-   .02 

.00 

.99 

.97 

0.99 

0.99 

.00 

4  .02 

14... 

4   .02 

.00 

.98 

1.00 

4-  .01 

-   .01 

15... 

+   .04 

.00 

.88 

.92 

4-  .11 

4  .07 

16... 

-    .02 

.00 

.96 

.94 

-   .04 

-   .01 

17... 

-   .06 

.00 

1.03 

.97 

-   .11 

-   .04 

18... 

4   .02 

.00 

.89 

.91 

.92 

.93 

4  .03 

4  .02 

19... 

+   .07 

.00 

.80 

.87 

4  .12 

4  .06 

20... 

+   -03 

.00 

.91 

.94 

4  .01 

-   .01 

21... 

-    .02 

.00 

.94 

.92 

-   .01 

4  .01 

22... 

-   .04 

.00 

.98 

.94 

-   .05 

-   .01 

23... 

4  -08 

.00 

.85 

.93 

.93 

.93 

4  .08 

.00 

24... 

4  .02 

—  .02 

.68 

(.68) 

4-  .25 

(4  .25) 

25... 

-    .21 

.00 

1.22 

(1.01) 

-   .29 

(-   .08) 

26... 

-    .06 

.00 

.97 

.91 

-   .12 

-    .02 

27... 

+   .02 

.00 

.86 

.88 

-   .01 

4  .01 

28... 

4   .12 

.00 

.77 

.89 

+  .08 

.00 

29... 

+   .03 

.00 

.84 

.87 

.85 

.89 

4  -01 

4  .02 

30... 

4   .04 

.00 

.84 

.88 

4  .01 

4   .01 

31... 

4  .08 

.00 

.83 

.91 

4  .02 

-   .02 

Aug.    1... 

+   .12 

.00 

.71 

.83 

4-  .09 

4  .02 

2... 

+   .03 

.00 

.84 

.87 

—  .04 

-   .02 

3... 

-   .02 

.00 

.89 

.87 

.80 

.85    • 

-   .09 

-   .02 

4... 

4-  .06 

.00 

.77 

.83 

+  .03 

4  .02 

5... 

+  .04 

.00 

.81 

.85 

-   .01 

.00 

6... 

4-  .01 

.00 

.85 

.86 

-   .02 

-   .04 

7... 

+  .10 

.00 

.72 

.82 

4-  .11 

.00 

8... 

-   .07 

.00 

.83 

.76 

.83 

.82    • 

.00 

4  .06 

9... 

-   .05 

.00 

.90 

.85 

-   .07 

-   .03 

10... 

-   .02 

.00 

.85 

.83 

-   .02 

-   .01 

11... 

-   .06 

.00 

.88 

.82 

-   .03 

4  .03 

12... 

4  .01 

.00 

.86 

.87 

-    .01 

-   .02 

13... 

4  .05 

.00 

.80 

.85 

.85 

.85 

4  .05 

.00 

14... 

4  .03 

.00 

.84 

.87 

4-  .01 

-   .02 

15... 

-  .04 

.00 

.89 

.85 

-  .04 

.00 

16... 

-    .04 

.00 

.94 

.90 

4-  .04 

+   .02 

17... 

-   .07 

.00 

.97 

.90 

4-  .01 

4  .02 

18... 

-   .11 

.00 

1.08 

.97 

.98 

.92 

-   .10 

-   .05 

19... 

-   .09 

.00 

1.01 

.92 

-  .03 

.00 

20... 

.00 

.00 

.91 

.91 

4  .07 

4  .01 

BAROMETRIC   PRESSURES   ON   THE   GREAT   LAKES  93 

TABLE  No.  22 — Continued. 


Date. 

Baro- 
metric 
cor- 
rection. 

Wind 
cor- 
rection. 

Observed 
eleva- 
tion 
579+. 

Cor- 
rected 
eleva- 
tion 
579+. 

5-day 
observed 
mean 
579+. 

5-day 
corrected 
mean 
579+. 

Residuals  from 
5-day  means. 

Obs. 

Cor. 

1911. 

feet. 

feet. 

feet. 

feet. 

feet. 

feet. 

feet. 

feet. 

Aug.  21... 

+0.08 

0.00 

0.83 

0.91 

+0.05 

-0.04 

22... 

-   .04 

.00 

.95 

.91 

-   .07 

-   .04 

23... 

-   .10 

.00 

.98 

.88 

0.88 

0.87 

-   .10 

-   .01 

24... 

.00 

.00 

.82 

.82 

+   .06 

+  .05 

25... 

+   .02 

.00 

.81 

.83 

+   .07 

+  .04 

26... 

+   .04 

.00 

.80 

.84 

-   .03 

-   .05 

27... 

+   .11 

.00 

.72 

.83 

+   .05 

-   .04 

28... 

-   .04 

.00 

.84 

.80 

-   .07 

-   .01 

29... 

-   .08 

.00 

.99 

(.91) 

.77 

.79 

-    .22 

(-   -12) 

30... 

+   .07 

.00 

.69 

.76 

+   .08 

+   .03 

31... 

+   .12 

.00 

.59 

.71 

+   .18 

+  .08 

Sept.    1... 

+   .02 

.00 

.65 

.67 

.00 

-   .02 

2... 

+   .01 

.00 

.71 

.72 

-   .06 

-   .07 

3... 

-   .13 

.00 

.81 

.68 

.65 

.65 

-   .16 

-   .03 

4... 

.00 

.00 

.60 

.60 

+   .05 

+   .05 

5... 

+   .09 

.00 

.49 

.58 

+   .16 

+  .07 

6... 

-   .05 

.00 

.66 

.61 

-   .10 

-   .01 

7... 

+   .06 

.00 

.54 

.60 

+   .02 

.00 

8... 

+   .07 

.00 

.44 

(.51) 

.56 

.60 

+  .12 

(+   .09) 

9... 

+   .05 

.00 

.56 

.61 

.00 

-   .01 

10... 

-   .01 

.00 

.60 

.59 

-   .04 

+   .01 

11... 

+   .05 

.00 

.57 

.62 

+   .03 

+  .02 

12... 

-   .07 

.00 

.84 

(.77) 

-   .24 

(-   -13) 

13... 

+   .05 

.00 

.65 

.70 

.60 

.64 

-   .05 

-   .06 

14... 

+   .21 

.00 

.43 

.64 

+  -17 

.00 

15... 

+   .10 

.00 

.49 

.59 

+   .11 

+   .05 

16... 

-   .03 

.00 

.67 

.64 

-   .08 

-   .03 

17... 

+   .07 

.00 

.52 

.59 

+   .07 

+   .02 

18... 

+  .17 

.00 

.43 

.60 

.59 

.61 

+   .16 

+   .01 

19... 

-   .07 

.00 

.70 

.63 

-   .11 

-   .02 

20... 

-   .03 

.00 

.64 

.61 

-   .05 

.00 

21... 

+   .06 

.00 

.50 

.56 

+   .03 

+  .03 

22... 

+   .06 

.00 

.57 

.63 

-   .04 

-   .04 

23... 

+  .13 

.00 

.49 

.62 

.53 

.59 

+  .04 

-   .03 

24... 

-   .02 

.00 

.58 

.56 

-   .05 

+   .03 

25... 

-   .05 

.00 

.49 

(.44) 

+  .04 

(+  .15) 

26... 

.00 

.00 

.57 

.57 

-   .03 

-   .01 

27... 

+  .02 

.00 

.51 

.53 

+  .03 

+   .03 

28... 

+  .03 

.00 

.57 

.60 

.54 

.56 

-   .03 

-   .04 

29... 

.00 

.00 

.45 

(.45) 

+   .09 

(+   .11) 

30... 

-   .02 

.00 

.58 

.56 

-   .04 

.00 

94 


EFFECTS   OF   WINDS   AND    OF 


TABLE  No.  23 — Observed  and  corrected  elevations  of  water  surface  at  Mackinaw  Gage  on 
Lake  Michigan-Huron. 


Date. 

Baro- 
metric 
cor- 
rection. 

Wind 
cor- 
rection. 

Observed 
eleva- 
tion 
579+. 

Cor- 
rected 
eleva- 
tion 
579+. 

5-day 
observed 
mean 
579+. 

5-day 
corrected 
mean 
579+. 

Residuals  from 
5-day  means. 

Obs. 

Cor. 

1911. 

feet. 

feet. 

feet. 

/*. 

feet. 

feet. 

feet. 

feet. 

June    1... 

-0.04 

0.00 

0.95 

0.91 

-0.06 

-0.02 

2... 

+   .05 

.00 

.87 

.92 

+   .02 

-   .03 

3... 

+   .02 

.00 

.85 

.87 

0.89 

0.89 

+   .04 

+   .02 

4... 

+   .01 

.00 

.87 

.88 

+   .02 

+   .01 

5... 

-   .02 

.00 

.90 

.88 

-   .01 

+   .01 

6... 

+  .03 

.00 

.86 

.89 

+   .08 

+   .02 

7... 

+   .02 

.00 

.91 

.93 

+   .03 

-   .02 

8... 

-   .02 

.00 

.92 

.90 

.94 

.91 

+   .02 

+   .01 

9... 

-   .09 

.00 

.97 

.88 

-   .03 

+   .03 

10... 

-   .10 

.00 

1.03 

.93 

-   .09 

-    .02 

11... 

-   .07 

.00 

1.06 

.99 

-   .03 

+  '.03 

12... 

-   .01 

.00 

.06 

.05 

-    .03 

-   .03 

13... 

.00 

.00 

.01 

.01 

1.03 

1.02 

+   .02 

+   .01 

14... 

+  .01 

.00 

.01 

.02 

+   .02 

.00 

15... 

.00 

.00 

.01 

.01 

+   .02 

+   .01 

16... 

-   .01 

.00 

.02 

.01 

.00 

+   .03 

17... 

+   .05 

.00 

.02 

.07 

.00 

-   .03 

18... 

+   .02 

.00 

.01 

.03 

1.02 

1.04 

+   .01 

+   .01 

19... 

-   .01 

.00 

.04 

.03 

-   .02 

+   .01 

20... 

+   .05 

.00 

.00 

.05 

+   .02 

-   .01 

21... 

-   .02 

.00 

.06 

.04 

-    .08 

-   .02 

22... 

-   .02 

.00 

.07 

.05 

-    .09 

-    .03 

23... 

+   .03 

.00 

.84 

(.87) 

.98 

1.02 

+    .14 

(+   -15) 

24... 

.00 

.00 

.94 

(.94) 

+   .04 

(+   .08) 

25... 

.00 

.00 

.98 

.98 

.00 

+   .04 

26... 

-   .06 

.00 

1.14 

(1.08) 

-    .12 

(-  .10) 

27... 

-   .02 

.00 

1.02 

1.00 

.00 

-   .02 

28... 

+   .05 

.00 

.94 

.99 

1.02 

.98 

+   .08 

-   .01 

29... 

+   .01 

.00 

.94 

.95 

+   .08 

+   .03 

30... 

-   .08 

.00 

1.04 

.96 

-    .02 

+   .02 

July     1... 

-   .07 

.00 

1.04 

.97 

-   .03 

+   .01 

2... 

-   .08 

.00 

1.08 

1.00 

-   .07 

-   .02 

3... 

+   .01 

.00 

1.00 

1.01 

1.01 

.98 

+   .01 

-   .03 

4... 

.00 

.00 

.96 

.96 

+   .05 

+   .02 

5... 

-   .05 

.00 

.99 

.94 

+   .02 

+   .04 

6... 

+   .07 

.00 

.94 

1.01 

+   .06 

-   .04 

7... 

+   .02 

.00 

.94 

.96 

+   .06 

+   .01 

8... 

-    .07 

.00 

1.02 

.95 

1.00 

.97 

-   .02 

+   .02 

9... 

-   .11 

.00 

1.06 

.95 

-   .06 

+   .02 

10... 

-   .09 

.00 

1.06 

.97 

-   .06 

.00 

BAROMETRIC   PRESSURES   ON   THE   GREAT   LAKES  95 

TABLE  No.  23 — Continued. 


Date. 

Baro- 
metric 
cor- 
rection. 

Wind 
cor- 
rection. 

Observed 
eleva- 
tion 
579+. 

Cor- 
rected 
eleva- 
tion 
579+. 

5-day 
sbserved 
mean 
579+. 

5-day 
orrected 
mean 
579+. 

Residuals  from 
5-day  means. 

Obs. 

Cor. 

1911. 

feet. 

feet. 

feet. 

feet. 

/•* 

{*. 

feet. 

feet. 

July  11... 

-0.05 

0.00 

1.03 

0.98 

-0.02 

-0.02 

12... 

-   .01 

.00 

1.00 

.99 

+   .01 

-   .03 

13... 

-   .04 

.00 

.99 

.95 

1.01 

0.96 

+   .02 

+   .01 

14... 

-   .04 

.00 

1.00 

.96 

+   .01 

.00 

15... 

-   .08 

.00 

1.01 

.93 

1 

.00 

+   .03 

16... 

+  .03 

.00 

.90 

.93 

+   .03 

-  .04 

17... 

-   .02 

.00 

.92 

.90 

+   .01 

-   .01 

18... 

-   .08 

.00 

.94 

.86 

.93 

.89 

-   .01 

+  .03 

19... 

-   .04 

.00 

.91 

.87 

+  .02 

+   .02 

20... 

-   .07 

.00 

.98 

.91 

-   .05 

-   .02 

21... 

-   .02 

.00 

.91 

.89 

+   .02 

-   .02 

22... 

-   .05 

.00 

.90 

.85 

+  .03 

+   .02 

23... 

-   .07 

.00 

.97 

.90 

.93 

.87 

-   .04 

-   .03 

24... 

-   .12 

.00 

1.01 

.89 

-   .08 

-   .02 

25... 

-   .02 

.00 

.85 

.83 

+  .08 

+  .04 

26... 

-   .03 

.00 

.90 

.87 

+   .02 

-   .01 

27... 

-   .07 

.00 

.92 

.85 

.00 

+  .01 

28... 

-   .10 

.00 

.94 

.84 

-   .02 

+  .02 

29... 

-   .06 

.00 

.91 

.85 

.92 

.86 

+   .01 

+   .01 

30... 

-   .07 

.00 

.93 

.86 

-   .01 

.00 

31... 

-   .06 

.00 

.93 

.87 

-   .01 

-   .01 

Aug.     1... 

-   .08 

.00 

.94 

.86 

-   .05 

.00 

2... 

-   .06 

.00 

.93 

.87 

-   .04 

-.01 

3... 

+   .04 

.00 

.80 

.84 

.89 

.86 

+  .09 

+  .02 

4... 

-   .03 

.00 

.88 

.85 

+  .01 

+  .01 

5.. 

-   .03 

.00 

.92 

.89 

-   .03 

-   .03 

6.. 

+   .01 

.00 

.88 

.89 

+   .01 

.00 

7.. 

-   .04 

.00 

.94 

.90 

-   .05 

-   .01 

8.. 

-   .03 

.00 

.92 

.89 

.89 

.89 

-   .03 

.00 

9.. 

-   .04 

.00 

.92 

.88 

-   .03 

+  .01 

10.. 

+   .09 

.00 

.78 

.87 

+   .11 

+   .02 

11.. 

+  .06 

.00 

.82 

.88 

+  .05 

.00 

12.. 

+   .03 

.00 

.86 

.89 

+  .01 

-   .01 

13.. 

-    .03 

.00 

.90 

.87 

.87 

.88 

-   .03 

+  .01 

14.. 

-   .02 

.00 

.89 

.87 

-   .02 

+  .01 

15.. 

+   .01 

.00 

.89 

.90 

-   .02 

—  .02 

16.. 

-   .02 

.00 

.92 

.90 

-   .05 

-   .02 

17.. 

+   .01 

.00 

.89 

.90 

-   .02 

-   .02 

18.. 

+   .06 

.00 

.80 

.86 

.87 

.88 

+   .07 

+  .02 

19.. 

+   .03 

.00 

.85 

.88 

+  .02 

.00 

20.. 

-   .02 

.00 

.90 

.88 

-   .03 

.00 

96 


EFFECTS   OF   WINDS   AND    OF 

TABLE  No  23— Continued. 


Date. 

Baro- 
metric 
cor- 
rection. 

Wind 
cor- 
rection. 

Observed 
eleva- 
tion 
579+. 

Cor- 
rected 
eleva- 
tion 
579+. 

5-day 
observed 
mean 
579+. 

5-day 
corrected 
mean 
579+. 

Residuals  from 
5-day  means. 

Obs. 

Cor. 

1911. 

feet. 

feet. 

feet. 

feet. 

feet. 

feet. 

feet. 

feet. 

Aug.  21... 

-0.11 

0.00 

0.98 

0.87 

-0.16 

-0.05 

22... 

+   .05 

.00 

.73 

(-78) 

+   .09 

(+    -04) 

23... 

+   .04 

.00 

.78 

.82 

0.82 

0.82 

+   .04 

.00 

24... 

+   .02 

.00 

.77 

.79 

+   .05 

+   .03 

25... 

-    .03 

.00 

.82 

.79 

.00 

+   .03 

26... 

-   .04 

.00 

.80 

.76 

-    .09 

-   .07 

27... 

-    .06 

.00 

.80 

.74 

-   .09 

-   .05 

28... 

+   .01 

.00 

.67 

.68 

+   .04 

+   .01 

29... 

+   .05 

.00 

.63 

.68 

.71 

.69 

+    .08 

+   .01 

30... 

.00 

.00 

.64 

.64 

+   .07 

+   .05 

31... 

-   .05 

.00 

.70 

.65 

+   .01 

+   .04 

Sept.    1... 

-   .05 

.00 

.71 

.66 

-    .02 

-    .01 

2... 

-    .02 

.00 

.64 

.62 

+   .05 

+   .03 

3... 

+   .03 

.00 

.68 

(.71) 

.69 

.65 

+   .01 

(-   -06) 

4... 

.00 

.00 

.67 

.67 

+   .02 

-    .02 

5... 

.00 

.00 

.73 

(.73) 

-    .04 

(-    -08) 

6... 

+   .08 

.00 

.53 

.61 

+   .04 

-    .01 

7... 

+   .08 

.00 

.49 

.57 

+   .08 

+   .03 

8... 

+   .03 

.00 

.58 

.61 

.57 

.60 

-    .01 

-   .01 

9... 

-    .02 

.00 

.64 

.62 

-   .07 

-   .02 

10... 

-   .03 

.00 

.63 

.60 

-    .06 

.00 

11... 

-    .05 

.00 

.65 

.60 

-   .03 

-   .02 

12... 

-f   .05 

.00 

.52 

.57 

+   .10 

+   .01 

13... 

+   .01 

.00 

.57 

.58 

.62 

.58 

+   .05 

.00 

14... 

-    .09 

.00 

.69 

.60 

-   .07 

-    .02 

15... 

-    .12 

.00 

.68 

.56 

-    .06 

+   .02 

16... 

-   .02 

.00 

.58 

.56 

+   .02 

+   .02 

17... 

-   .01 

.00 

.63 

.62 

-    .03 

-   .04 

18... 

-    .09 

.00 

.69 

.60 

.60 

.58 

-   .09 

-    .02 

19... 

+   .01 

.00 

.56 

.57 

+   .04 

+   .01 

20... 

+   .01 

.00 

.56 

.57 

+   .04 

+   .01 

21... 

+   .03 

.00 

.55 

.58 

+   .02 

-    .02 

22... 

-    .10 

.00 

.64 

.54 

-   .07 

+   .02 

23... 

-   .11 

.00 

.71 

.60 

.57 

.56 

-   .14 

-   .04 

24... 

+   .07 

.00 

.46 

.53 

+   .11 

+   .03 

25... 

+  .06 

.00 

.51 

.57 

J 

+   .06 

-   .01 

26... 

-    .03 

.00 

.57 

.54 

-    .07 

-    .03 

27... 

.00 

.00 

.52 

.52 

-   .02 

-   .01 

28... 

.00 

.00 

.46 

.46 

.50 

.51 

+   .04 

+   .05 

29... 

+   .04 

.00 

.48 

.52 

+   .02 

-   .01 

30... 

.00 

.00 

.49 

.49 

+   .01 

+   .02 

BAROMETRIC   PRESSURES   ON   THE   GREAT   LAKES 


97 


TABLE  No.  24 — Observed  and  corrected  elevations  for  the  whole  surface  of  Lake  Erie  at 
derived  from  Buffalo  and  Cleveland  observations  combined. 


Date. 

Observed 
elevation 
570+. 

Corrected 
elevation 
570  +. 

5-day 
observed 
mean 
570-K 

5-day 
corrected 
mean 
570  +. 

Residuals  from 
5-day  means. 

Obs. 

Cor. 

1910. 
June     1  
2  
3  

feet. 
2.76 
2.59 
2.64 
2.48 
2.56 

2.64 
2.68 
2.64 
2.54 
2.49 

2.52 
2.68 
2.63 
2.62 
2.63 

2.59 
2.62 
2.64 
2.61 
2.58 

2.58 
2.56 
2.57 
2.20 
2.39 

2.41 
2.59 
2.60 

2.48 
2.48 

2.48 
2.44 
2.46 
2.29 
2.26 

2.33 
2.40 
2.38 
2.38 
2.39 

feet. 
2.60 
2.63 
2.60 
2.58 
2.59 

2.56 
2.56 
2.59 
2.62 
2.60 

2.60 
2.56 
2.59 
2.58 
2.60 

2.59 
2.56 
2.57 
2.60 
2.58 

2.59 
2.56 
2.56 
2.47 
2.60 

2.40 
2.36 
2.50 
2.43 
2.45 

2.40 
2.41 
2.39 
2.36 
2.36 

2.38 
2.36 
2.38 
2.35 
2.33 

feet. 
2.61 

2,60 
2.62 
2.61 
2.46 
2.51 
2.39 
2.38 

feet. 
2.60 

2.59 
2.59 
2.58 
2.56 
2.43 
2.38 
2.36 

feet. 
-0.15 
+   .02 
-   .03 
+   .13 
+   .05 

-    .04 
-   .08 
-   .04 
+   .06 
+   .11 

-{-   .10 
-   .06 
-   .01 
.00 
-   .01 

+  .02 
-   .01 
-   .03 
.00 
+   .03 

-   .12 
-   .10 
-   .11 

+   .26 
+   .07 

+  .10 
-   .08 
-   .09 
+   .03 
+  .03 

-  .09 
-   .05 
-   .07 
+  .10 
+  .13 

+  .05 
—  .02 
.00 
.00 
—  .01 

feet. 
0.00 
-   .03 
.00 
-1-   .02 
-I-  .01 

+  .03 
+   .03 
.00 
-   .03 
-   .01 

-   .01 
+   .03 
.00 
+   .01 
-   .01 

-   .01 
+   .02 
+   .01 
-   .02 
.00 

-   .03 
.00 
.00 
+   .09 
-   .04 

+  .03 
+  .07 
-   .07 
.00 
-   .02 

-   .02 
-   .03 
-   .01 
+  .02 
+   .02 

—  .02 
.00 
—  .02 
+   .01 
+  .03 

4    

5 

6  

7  

8  

9    

10  

11  

12  

13  
14  

15  

16  
17  

18  
19  

20 

21      

22 

23 

24  

25   .      ... 

26  
27 

28  
29  
30  

July     1 

2  

3    .    .  .  . 

4  

5  

6  
7  
8  

9   

10  

98 


EFFECTS   OF   WINDS   AND   OF 

TABLE  No.  24— Continued. 


Date. 

Observed 
elevation 
570+. 

Corrected 
elevation 
570+. 

5-day 
observed 
mean 
570  -K 

5-day 
corrected 
mean 
570  +. 

Residuals  from 
5-day  means. 

Obs. 

Cor. 

1910. 
July    11  

feet. 
2.42 
2.36 
2.50 
2.44 
2.45 

2.38 
2.30 
2.28 
2.33 
2.29 

2.34 
2.38 
2.30 
2.42 
2.52 

2.44 
2.41 
2.38 
2.38 
2.46 
2.38 

2.36 
2.34 
2.30 
2.52 
2.56 

2.36 
2.20 
2.20 
2.18 
2.29 

2.26 
2.15 
2.08 
2.10 
2.14 

2.12 
2.11 
2.20 
2.17 
2.12 

feet. 
2.32 
2.37 
2.41 
2.46 
2.45 

2.17 

2.09 
2.35 
2.34 
2.20 

2.29 
2.31 
2.32 
2.20 
2.13 

2.39 
2.38 
2.36 
2.37 
2.40 
2.31 

2.36 
2.37 
2.34 
2.37 
2.31 

2.29 
2.25 
2.24 
2.24 
2.26 

2.26 
2.21 
2.04 
2.07 
2.20 

2.21 
2.21 
2.22 
2.20 
2.19 

feet. 
2.43 

2.32 
2.39 

2.41 

2.42 
2.25 
2.15 
2.14 

feet. 
2.40 

2.23 
2.25 

2.37 

2.35 
2.26 
2.16 
2.21 

feet. 
+0.01 
+  .07 
-   .07 
-   .01 
-   .02 

-   .06 
+   .02 
+   .04 
-   .01 
+  .03 

+  .05 
+  .01 
+  .09 
-   .03 
-   .13 

-    .03 
.00 
+   .03 
+   .03 
-    .05 
+   .03 

+   .06 
+   .08 
+   .12 
-.   10 
-   .14 

-    .11 
+   .05 
+   .05 
+   .07 
-   .04 

-   .11 
.00 
+  .07 
+  .05 
+  .01 

+  .02 
+  .03 
-   .06 
-  .03 
+   .02 

feet. 
+0.08 
+  .03 
-   .01 
-   .06 
-   .05 

+  .06 
+  .14 
-   .12 
-   .11 
+  .03 

-   .04 
-   .06 
-   .07 
+  .05 
+  .12 

-    .02 
-   .01 
+   .01 
.00 
-   .03 
+   .06 

-   .01 
-   .02 
+   .01 
-   .02 
+   .04 

-   .03 
+   .01 
+   .02 
+   .02 
.00 

-   .10 
-   .05 
+   .12 
+  .09 
-   .04 

.00 
.00 
-   .01 
+  .01 
+   .02 

12  
13  

14  
15  

16  

17  
18  

19  

20 

21  
22  

23  
24  

25 

26  
27  
28  
29  

30  
31  

Aug     1 

2  

3  

4  

5 

6 

7  

8  , 
9  

10.    . 

11  

12  

13  

14 

15  

16  

17.  . 

18  

19.  .    . 

20  

BAROMETRIC   PRESSURES   ON   THE    GREAT   LAKES  99 

TABLE  No.  24 — Continued. 


Date. 

Observed 
elevation 
570+. 

Corrected 
elevation 
570+. 

5-day 
observed 
mean 
570+. 

5-day 
corrected 
mean 
570  +. 

Residuals  from 
5-day  means. 

Obs. 

Cor. 

1910. 
Aug   21 

feet. 
2.07 
2.10 
2.06 
2.06 
2.24 

2.21 

2.04 
1.96 
1.77 
1.90 
2.00 

1.98 
2.02 
2.05 
2.20 
2.04 

2.23 
2.10 
2.06 
2.15 
1.95 

1.96 
1.97 
1.92 
1.90 
1.90 

1.71 
1.80 
1.88 
1.75 
1.90 

1.86 
1.66 
1.78 
1.71 
1.92 

1.52 
1.98 
1.86 
1.74 
1.80 

feet. 
2.16 
2.15 
2.10 
2.10 
2.11 

2.15 
2.03 
2.03 
2.18 
2.05 
2.02 

2.01 
2.11 
2.14 
2.19 
2.10 

2.13 

2.03 
2.09 
2.06 
1.99 

2.01 
1.98 
1.91 
1.89 
1.89 

1.76 
1.73 

1.88 
2.01 
1.89 

1.87 
1.98 
1.87 
1.85 
1.85 

1.62 
1.92 
1.82 
1.78 
1.79 

feet. 
2.11 

1.98 

2.06 
2.10 
1.93 
1.81 
1.79 
1.78 

feet. 
2.12 

2.08 

2.11 
2.06 
1.94 
1.85 

1.88 
1.79 

feet. 
+0.04 
+   .01 
+   .05 
+   .05 
-   .13 

-   .23 
-   .06 
+   .02 
+   .21 
+   .08 
-   .02 

+  .08 
+  .04 
+  .01 
-   .14 
+   .02 

-   .13 
.00 
+   .04 
-   .05 
+   .15 

-   .03 
-   .04 
+  .01 
+  .03 
+  .03 

+  .10 
+  .01 
-   .07 
+   .06 
-   .09 

-   .07 
+   .13 
+   .01 
+   .08 
-   .13 

+   .26 
-   .20 
-   .08 
+   .04 
-   .02 

feet. 
-0.04 
-   .03 
+   .02 
+  .02 
+   .01 

-   .07 

+   .05 
+   .05 
-   .10 

+  .03 
+   .06 

+  .10 
.00 
-  .03 
-   .08 
+   .01 

-   .07 
+   .03 
-   .03 
.00 
+  .07 

-   .07 
-  .04 
+   .03 
+   .05 
+  .05 

+  .09 
+  .12 
-   .03 
-   .16 
-   .04 

+   .01 
-   .10 
+   .01 
+   .03 
+   .03 

+  .17 
-   .13 
-   .03 
+   .01 
.00 

22  
23  
24  
25  

26  

27 

28  
29  
30  

31  

Sept.    1  
2  
3  
4  
5  

6  

7    

8  
9  
10 

11    . 

12 

13  

14  

15  

16  

17 

18  
19  
20  

21  
22  
23    . 

24  

25  
26 

27  

28  
29  
30  

100 


EFFECTS   OF   WINDS  AND    OF 

TABLE  No.  24 — Continued. 


Date. 

Observed 
elevation 
570+. 

Corrected 
elevation 
570+. 

5-day 
observed 
mean 
570  +. 

5-day 
corrected 
mean 
570  +. 

Residuals  from 
5-day  means. 

Obs. 

Cor. 

1910. 
Oct.     1  
2   .  .      . 

feet. 
2.10 
1.70 
1.68 
1.79 
1.76 

2.07 
1.90 
1.93 
2.08 
1.94 

2.05 
.80 
.50 
.90 
.94 

.93 

.81 
.85 
.78 
.76 

.72 
.34 

.98 
.84 
.89 

1.77 
2.18 
1.85 
1.88 
1.84 
1.72 

feet. 
1.87 
1.77 
1.78 
1.76 
1.96 

1.73 
1.92 
1.93 
1.94 
1.88 

1.83 
1.86 
1.71 
1.84 
1.85 

1.80 
1.85 
1.86 
1.69 
1.82 

2.01 

i'.87 
1.76 
1.81 

1.78 
1.80 
1.86 
1.85 

i!si 

feet. 
1.81 

1.98 
1.84 
1.83 
1.95 

1.87 

feet. 
1.83 

1.88 
1.82 
1.80 
1.86 

1.82 

feet. 
-0.29 
+   .11 
+   .13 
+   .02 
+   .05 

-    .09 
+   .08 
+   .05 
-   .10 
+   .04 

-   .21 
+   .04 
+   .34 
-   .06 
-   .10 

-   .10 
+   .02 
-   .02 
+   .05 
+   .07 

+   .23 
-   .39 
-   .03 
+   .11 
+   .06 

+  .10 

-   .31 

+   .02 
-   .01 
+   .03 
+   .15 

feet. 
-0.04 
+   .06 
+   .05 
+   .07 
-   .13 

+   .15 
-   .04 
-   .05 
-   .06 
.00 

-   .01 
-   .04 
+   .11 
-   .02 
-    .03 

.00 
-   .05 
-   .06 
+   -11 
-    .02 

-   .15 

-   .01 
+   .10 
+   .05 

+   .04 
+  .02 
-   .04 
-   .03 

+  '.'6l 

3  

4  

5 

6 

7  
8  
9  

10  

11  

12 

13  

14 

15  

16  

17  
18  

19  
20  

21  
22  

23  
24  

25  

26  
27  

28  

29  

30  
31 

In  table  No.  24  the  corrected  elevation  for  any  day  is  the  weighted  mean 
of  the  corrected  elevation  for  Buffalo  as  recorded  in  table  No.  19  and  the 
corrected  elevation  for  Cleveland  as  recorded  in  table  No.  20,  the  relative 
weights  assigned  being  1.0  for  Buffalo  and  1.7  for  Cleveland. 

The  general  explanations  given  on  pages  78-79  with  reference  to  5-day 
means  and  residuals  in  table  No.  19  also  apply  to  table  No.  24. 

MEAN  ELEVATIONS  OF  LAKE  MICHIGAN-HURON. 

The  general  explanations  made  in  connection  with  table  No.  24  for  Lake 
Erie  apply  to  table  No.  25  for  Lake  Michigan-Huron. 


BAROMETRIC   PRESSURES   ON   THE    GREAT  LAKES 


101 


TABLE  No.  25 — Observed  and  corrected  elevations  for  the  whole  surface  of  Lake  Michigai 
Huron  as  derived  from  Milwaukee,  Harbor  Beach,  and  Mackinaw  observations  combined. 


Date. 

Observed 
elevation 
579+. 

Corrected 
elevation 
579+. 

5-day 
observed 
mean 
579+. 

5-day 
corrected 
mean 
579+. 

Residuals  from 
5-day  means. 

Obs. 

Cor. 

1911. 

feet. 
0.92 
.91 
.92 
.95 
.96 

.97 
.96 
.99 
.98 
.98 

1.00 
1.02 
1.03 
1.03 
1.03 

1.04 
1.06 
1.04 
1.03 
1.05 

1.05 
1.04 
1.01 
1.01 
1.03 

1.05 
1.00 
1.02 
1.02 
1.01 

1.00 
1.00 
1.01 
1.00 
.99 

1.02 
1.03 
1.01 
.97 
.98 

feet. 
0.93 
.92 
.90 
.92 
.93 

.94 
.94 
.96 
.95 
.97 

1.01 

1.04 
1.02 
1.03 
1.02 

1.04 
1.06 
1.03 
1.04 
1.04 

1.04 
1.05 
1.06 
1.10 
1.01 

1.02 
1.00 
1.01 
.99 
.99 

1.00 
1.01 
1.01 
.99 

.98 

1.02 
1.00 
1.00 
.99 
.97 

feet. 

[    0.93 

1 

.98 
1.02 
1.04 
1.03 
1.02 
1.00 
1.00 

feet. 
0.92 

.95 
1.02 
1.04 
1.05 
1.00 
1.00 
1.00 

feet. 
+0.01 
+   .02 
+   .01 
-   .02 
-    .03 

+   .01 
+   .02 
-   .01 
.00 
.00 

+   .02 
.00 
-   .01 
-   .01 
-   .01 

.00 
-   .02 
.00 
+   .01 
-   .01 

-   .02 
-   .01 
+  .02 
+  .02 
.00 

-   .03 
+   .02 
.00 
.00 
+   .01 

.00 
.00 
-   .01 
.00 
+   .01 

-   .02 
-   .03 
-   .01 
+   .03 
+   .02 

feet. 
-0.01 
.00 
+   .02 
.00 
-   .01 

+   .01 
+   .01 
-   .01 
.00 
-   .02 

+  .01 
-   .02 
.00 
-   .01 
.00 

.00 
-   .02 
+   .01 
.00 
.00 

+  .01 
.00 
-   .01 
-   .05 
+  .04 

-   .02 
.00 
-   .01 
+   .01 
+   .01 

.00 
-   .01 
-   .01 
+   .01 
+   .02 

-   .02 
.00 
.00 
+   .01 
+   .03 

2  

3 

4  

5  
6     .  .      . 

7  

g 

9  

10  
11  

12....... 
13  

14  
15  

16  
17   . 

18  

19  

20 

21  
22  
23   . 

24  
25  

26  

27 

28  

29 

30  

July     1 

2  

3  
4  
5.  .  .     . 

6  

7 

8  
9   

10  

102 


EFFECTS   OF   WINDS   AND    OF 
TABLE  No.  25 — Continued. 


Date. 

Observed 
elevation 
579+. 

Corrected 
elevation 
579+. 

5-day 
observed 
mean 
579+. 

5-day 
corrected 
mean 
579+. 

Residuals  from 
5-day  means. 

Obs. 

Cor. 

1911. 
July    11  
12  

feet. 
0.96 
.96 
.95 
.96 
.94 

.94 
.94 
.91 
.89 
.91 

.87 
.88 
.88 
.78 
.85 

.87 
.86 
.85 
.85 
.85 
.86 

.85 
.87 
.85 
.86 
.86 

.87 
.87 
.85 
.86 
.90 

.92 
.89 
.89 
.89 

.88 

.90 
.91 
.90 
.91 
.89 

feet. 
1.00 
.99 
.97 
.98 
.95 

.94 
.94 
.91 
.90 
.93 

.90 
.89 
.90 
.88 
.82 

.88 
.86 
.86 
.86 
.86 
.87 

.86 
.87 
.84 
.85 
.87 

.87 
.87 
.86 
.87 
.88 

.89 
.88 
.88 
.89 
.89 

.91 
.92 
.90 
.90 
.89 

feet. 
0.95 

.92 

.85 

.86 

.86 
.87 
.89 
.90 

feet. 
0.98 

.92 

.88 

.86 

.86 
.87 
.89 
.90 

feet. 
-0.01 
-   .01 
.00 
-   .01 
+   .01 

-   .02 
-   .02 
+   .01 
+   .03 
+  .01 

-   .02 
-   .03 
-   .03 
+   .07 
.00 

-   .01 
.00 
+  .01 
+  .01 
+  .01 
.00 

+  .01 
-   .01 
+   .01 
.00 
.00 

.00 
.00 
+  .02 
+   .01 
-   .03 

-   .03 
.00 
.00 
.00 
+  .01 

.00 
-  .01 
.00 
-   .01 
+   .01 

feet. 
-0.02 
-    .01 
+  .01 
.00 
+  .03 

-   .02 
-   .02 
+   .01 
+   .02 
-    .01 

-   .02 
-   .01 
-   .02 
.00 
+  .06 

-   .02 
.00 
.00 
.00 
.00 
-   .01 

.00 
-   .01 
+  .02 
+   .01 
-   .01 

.00 
.00 
+  .01 
.00 
-   .01 

.00 
+  .01 
+  .01 
.00 
.00 

-  .01 
-   .02 
.00 
.00 
+  .01 

13 

14  
15 

16  . 

17  

18  

19 

20  
21  

22  
23  

24 

25  

26  

27 

28  
29   . 

30  
31   . 

Aug.    1  
2  

3   .     .   . 

4  

5     .      .. 

6  
7  
8  
9 

10  

11  

12 

13  

14   . 

15  
16  

17  

18  

19  

20 

BAROMETRIC   PRESSURES   ON   THE   GREAT  LAKES  103 

TABLE  No.  25 — Continued. 


Date. 

Observed 
elevation 
579+. 

Corrected 
elevation 
579+. 

5-day 
observed 
mean 
579+. 

5-day 
corrected 
mean 
579+. 

Residuals  from 
5-day  means. 

Obs. 

Cor. 

1911. 
Aug    21 

feet. 
0.85 
.84 
.84 
.81 
.81 

.79 
.76 
.73 
.77 
.69 
.69 

.68 
.67 
.72 
.68 
.67 

.70 
.70 
.67 
.67 
.66 

.64 
.70 
.66 
.61 
.59 

.61 
.66 
.64 
.62 
.63 

.61 
.62 
.61 
.62 
.60 

.60 
.57 
.59 
.60 

.58 

feet. 
0.87 
.89 
.84 
.80 
.81 

.79 
.77 
.73 
.70 
.68 
.68 

.68 
.68 
.70 
.66 
.64 

.66 
.64 
.68 
.66 
.65 

.64 
.63 
.64 
.62 
.60 

.61 
.64 
.63 
.61 
.61 

.60 
.60 
.62 
.59 
.62 

.57 
.56 
.55 
.52 
.55 

feet. 
0.83 

.74 

.68 
.68 
.64 
.63 
.61 
.59 

feet. 
0.84 

.72 

.67 
.66 
.63 
.62 
.61 
.55 

feet. 
-0.02 
-   .01 
-   .01 
+  .02 
+   .02 

-   .05 
-   .02 
+   .01 
-   .03 
+   .05 
+   .05 

.00 
+  .01 
-   .04 
.00 
+   .01 

-   .02 
-   .02 
+  .01 
+   .01 

+   .02 

.00 
-   .06 
-   .02 
+  .03 
+   .05 

+   .02 
-   .03 
-   .01 
+   .01 
.00 

.00 
-   .01 
.00 
-   .01 
+  .01 

-   .01 
+  .02 
.00 
-   .01 
+   .01 

feet. 
-0.03 
-   .05 
.00 
+  .04 
+  .03 

-   .07 
-   .05 
-   .01 
+   .02 
+   .04 
+   .04 

-   .01 

-   .01 
-   .03 
+   .01 
+   .03 

.00 
+   .02 
-   .02 
.00 
+  .01 

-   .01 
.00 
-   .01 
+   .01 
+   .03 

+   .01 
-   .02 
-   .01 
+   .01 
+   .01 

+   .01 
+  .01 
-   .01 
+   .02 
-   .01 

-   .02 
-   .01 
.00 
+   .03 
.00 

22  

23   .      . 

24  

25   

26   

27 

28  

29 

30  

31 

Sept     1   .... 

2  

3   

4  

5  

6  

7  .     .. 

8  

9   

10  

11  
12  

13 

14  
15 

16.. 

17  

18  

19  
20  

21 

22  
23  
24  

25.  .      .. 

26  
27  
28  
29  

30 

104  EFFECTS  OF  WINDS  AND  OF 

In  table  No.  25  the  observed  elevation  for  any  day  was  obtained  by 
merely  taking  a  mean  of  the  three  observed  elevations  for  that  day  as  re- 
corded in  table  No.  21  for  Milwaukee,  in  table  No.  22  for  Harbor  Beach, 
and  in  table  No.  23  for  Mackinaw. 

In  table  No.  25  the  corrected  elevation  for  any  day  is  the  weighted  mean  of 
the  three  corrected  elevations  as  given  in  tables  No.  21,  22,  and  23,  the  rela- 
tive weights  assigned  being  1.9  for  Milwaukee,  2.9  for  Harbor  Beach,  and  5.1 
for  Mackinaw. 

EXPLANATION  OF  PLATES  7  to  13. 

Plates  7  to  13,  inclusive,  show  graphically  a  considerable  part  of  the  in- 
formation which  is  given  in  numerical  form  in  tables  Nos.  19  to  25,  inclusive. 
These  graphs  are  especially  valuable  as  a  means  of  showing  the  increase  in  ac- 
curacy which  has  been  secured  by  applying  corrections  for  wind  effects  and 
barometric  effects.  The  desired  values  are  the  elevations  of  the  mean  sur- 
face of  the  whole  of  a  lake  from  day  to  day.  Consider  the  evidence  in  these 
graphs  that  the  corrected  elevations  are  much  closer  approximations  to  the 
desired  elevations  of  the  mean  surface  of  the  whole  lake  from  day  to  day 
than  are  the  directly  observed  elevations. 

Plates  7,  8,  9,  and  10  combined  show  continuous  graphs  for  Lake  Erie 
from  June  1,  1910,  to  October  31,  1910.  The  graphs  on  the  lower  half  of 
each  of  the  plates  7,  8,  and  9  are  continuations  of  those  on  the  upper  half  of 
the  same  plate,  and  are  continued,  in  turn,  by  those  on  the  upper  half  of 
the  next  plate.  Plate  10  is  a  continuation  from  plate  9  on  the  same  scale. 
On  account  of  the  extreme  fluctuations  in  observed  elevations  in  the  last 
10  days  of  October,  it  was  necessary  to  use  the  whole  of  plate  10  for  a 
single  set  of  graphs. 

Similarly,  plates  11  to  13  combined  show  continuous  graphs  for  Lake 
Michigan-Huron  from  June  1,  1911,  to  September  30,  1911. 

Throughout  these  plates  7  to  13  the  elevations  directly  observed  at  each 
gage,  or  means  of  such  elevations  from  the  two  or  three  gages  on  each  lake, 
are  indicated  by  a  dash-and-dot  line — a  long  dash  and  a  dot  alternately. 

The  corrected  elevations  from  each  gage,  from  the  weighted  mean  of  the 
two  or  three  gages  on  a  lake,  are  indicated  by  a  continuous  solid  line. 

The  difference  between  the  dash-and-dot  line  and  the  continuous  line  has 
been  produced  by  the  application  of  corrections  for  wind  effects  and  baro- 
metric effects.  The  question  on  which  it  is  desired  to  concentrate  attention 
at  present  is,  How  much  more  accurately  in  each  case  does  the  continuous 
line  represent  the  fluctuation  in  elevation  of  the  mean  lake  surface  than  the 
dash-and-dot  line? 

The  figures  showing  elevations,  at  the  right-hand  margin  of  each  plate, 
are  feet  and  refer  to  mean  sea-level  as  a  datum. 

The  computed  fluctuation  in  elevation  of  the  mean  lake  surface,  in  each 
case,  as  produced  by  the  rainfall  on  the  lake  surface,  the  inflow  into  the  lake 
from  the  next  lake  above  in  the  chain  of  Great  Lakes,  and  the  outflow  to  the 


BAROMETRIC  PRESSURES  ON  THE  GREAT  LAKES     105 

next  lake  below  in  the  chain,  is  shown  for  each  lake  by  a  line  made  up  of  suc- 
cessive groups  of  two  short  dashes  and  a  dot.  This  line  is  plotted  from  the 
measured  values  of  the  rainfall  on  the  lake  surface,  as  determined  by  rain 
gages  around  the  lake,  and  from  the  measured  inflow  and  outflow  in  the 
streams  connecting  the  chain  of  Great  Lakes.  Note  the  smoothness  of  this 
line.  In  general,  it  rises  or  falls  less  than  0.02  foot  in  a  day,  usually  not 
more  than  0.01  foot.  In  other  words,  the  fluctuation  of  the  elevation  of  the 
mean  surface  of  the  lake  due  to  rainfall,  inflow,  and  outflow  is  normally  at 
the  very  slow  rates  stated.  The  extreme  case  anywhere  on  these  graphs  is 
the  rise  of  0.19  foot  in  3  days,  October  4-7,  1910,  of  Lake  Erie,  due  to  ex- 
tremely heavy  rainfall  over  the  whole  lake  surface  on  those  days. 

ACCURACY  AS  TESTED  BY  GRAPHS. 

The  mean  surface  of  the  whole  of  any  one  of  the  Great  Lakes  changes 
only  as  the  total  content  of  the  lake  changes.  That  total  content  changes 
from  five  causes  only:  (1)  rainfall  on  the  lake  surface,  (2)  inflow  from  the 
next  lake  above,  (3)  outflow  to  the  next  lake  below,  (4)  run-off  into  the 
lake  from  the  surrounding  land-drainage  area,  and  (5)  evaporation  from  the 
lake  surface. 

The  dot-and-two-dashes  graph,  commented  upon  above  and  based  upon 
direct  measurement,  shows  how  slow  and  regular  is  the  variation  of  elevation 
of  the  mean  lake  surface  due  to  the  first  three  of  the  causes  enumerated  in 
the  preceding  paragraph.  It  is  believed  that  the  variation  in  elevation  due 
to  the  fourth  and  fifth  causes — run-off  and  evaporation — is  even  smaller  and 
more  regular  than  that  due  to  the  first  three.  The  run-off  and  the  evapo- 
ration in  question  have  not  been  measured  directly.  From  sources  of  in- 
formation which  are  in  part  external  to  this  investigation,  it  is  estimated 
that  during  the  months  June  to  October  of  each  year  the  run-off  into  Lake 
Erie  from  the  surrounding  land-drainage  area  is  such  as  to  produce  a  rise 
from  0.004  to  0.040  foot  per  day  in  the  mean  lake  surface,  with  only  a  small 
percentage  of  days  in  which  the  rise  is  more  than  0.020  foot.  For  Lake 
Michigan-Huron  the  run-off  expressed  in  the  same  terms  is  even  more  con- 
stant. So,  too,  from  external  evidence,  it  is  estimated  that  on  either  lake 
during  the  season  June  to  October  the  evaporation  produces  a  fall  in  the 
mean  lake  surface  varying  from  but  little  more  than  0.000  foot  on  some  days 
to  0.021  foot  on  days  of  extremely  rapid  evaporation.  The  considerations 
indicated  in  this  paragraph  lead  to  the  belief  that  the  actual  variation  in  the 
elevation  in  the  mean  surface  of  either  Lake  Erie  or  Lake  Michigan-Huron 
is  so  slow  and  regular  as  to  be  properly  represented  by  a  graph  but  slightly 
less  smooth  and  regular  than  the  dot-and-two-dashes  graph  shown  in  plates 
7  to  13.  It  is  believed  that  the  actual  variation  of  the  mean  elevation  of  the 
whole  surface  of  either  lake  is  as  a  rule  about  0.01  foot  or  less  in  a  day,  is 
sometimes  somewhat  more  that  0.02  foot  in  a  day,  and  only  on  very  rare 
occasions  exceeds  0.08  foot  in  any  one  day. 


106  EFFECTS  OF  WINDS  AND  OF 

Hence,  apparent  fluctuations  in  the  graphs  of  observed  elevations  or  of 
corrected  elevations  which  exceed  the  rates  stated  in  the  last  sentence  of  the 
preceding  paragraph  are  evidence  that  said  graphs  are  imperfect  represen- 
tations of  the  rise  and  fall  of  the  mean  surface  of  the  lake.  The  nearer  a 
given  graph  approaches  to  the  smoothness  of  the  dot-and-two-dashes  graph, 
or  to  the  rates  of  fluctuations  stated  in  the  last  sentence  of  the  preceding 
paragraph,  the  more  accurately  does  it  represent  the  fluctuation  of  the  mean 
surface  of  the  lake.  If  one  examines  the  graphs  of  observed  elevations  and 
of  corrected  elevations  on  plates  7  to  13  with  this  point  of  view,  it  becomes 
clearly  evident  that  the  corrected  elevations  are  much  more  accurate  than 
the  observed  elevations. 

Examine  first  the  Buffalo  graphs  on  plates  7  to  10. 

Even  for  June  1  to  July  15,  1910,  a  comparatively  quiet  part  of  the  year, 
when  the  wind  effects  and  barometric  effects  at  Buffalo  are  much  smaller 
than  in  October,  note  how  much  smoother  is  the  continuous  (corrected 
elevation)  graph  than  the  dot-and-one-dash  (observed  elevation)  graph  on 
plate  7.  Note,  for  example,  the  following  contrast:  On  June  1-4  the  ob- 
served elevation  decreased  0.67  foot  in  three  days,  immediately  increased 
0.50  foot  in  the  three  days  June  4-7,  then  decreased  0.51  foot  in  the  three  days 
June  7-10,  and  then  increased  0.52  foot  in  the  two  days  June  10-12.  The 
changes  in  the  corrected  elevations  in  these  same  successive  periods  were 
respectively  a  decrease  of  0.12  foot,  an  increase  of  0.11  foot,  no  change,  and  a 
decrease  of  0.15  foot.  The  fluctuations  in  the  corrected  graph  were  not  more 
than  one-fourth  as  rapid  as  on  the  observed  graph  in  the  interval  June  1-12. 
During  the  period  June  1-July  15  the  greatest  change  in  the  observed  ele- 
vation (dot-and-dash  graph)  in  any  single  day  was  0.50  foot  decrease  on 
July  3-4.  During  the  same  period  as  covered  by  plate  7  the  greatest  change 
in  a  single  day  in  the  corrected  elevation,  as  shown  by  the  continuous  graph, 
was  a  decrease  of  0.20  foot  June  25-26.  It  is  evident  that  the  corrected 
elevations  at  Buffalo  do  not  represent  the  fluctuations  in  the  mean  surface  of 
Lake  Erie  without  error,  for  there  are  days  on  which  the  corrected  elevation 
changes  more  rapidly  than  it  is  possible  for  the  elevation  of  the  mean  surface 
of  Lake  Erie  to  change.  But  it  is  equally  evident  from  the  graphs  that  the 
corrected  elevations  are  a  much  more  accurate  representation  of  the  fluctu- 
ations of  the  mean  surface  than  are  the  observed  elevations.  .From  an 
inspection  of  plate  7,  one  may  conclude  that  the  errors  in  the  corrected  ele- 
vations are  from  one-half  to  one-quarter  as  large  as  those  in  the  observed 
elevations. 

Consider  next  the  Buffalo  graphs  on  plate  8.  In  general,  one  finds  the 
same  kind  of  contrasts  as  on  plate  7.  Note,  for  example,  the  range  of  0.69 
foot  in  the  observed  elevations  in  the  period  July  17-25,  1910.  In  the  same 
period  the  total  range  of  the  corrected  elevations  was  only  0.21  foot.  Note, 
also,  the  extremely  high  points  on  the  dot-and-dash  graph  on  August  5  and 
August  25  and  the  relative  smoothness  of  the  continuous  graph  near  those 
points. 


BAROMETKIC    PRESSURES   ON   THE   GREAT  LAKES  107 

On  plate  9,  examine  the  Buffalo  graphs  for  the  period  September  18-Octo- 
ber  4.  Note  the  extreme  irregularity  of  the  dot-and-dash  graph  (observed 
elevations),  showing  an  extreme  change  of  0.87  foot  on  October  1-2,  and 
the  relative  smoothness  of  the  continuous  graph  (corrected  elevations),  with 
no  change  in  a  single  day  of  more  than  0.12  foot. 

Plate  10  covers  a  stormy  period,  October  10-31,  1910.  Note  the  extreme 
irregularity  of  the  observed  elevations  at  Buffalo.  On  October  21-22  the 
observed  elevation  increased  1.75  feet  in  a  single  day.  The  corrected  ele- 
vation shows  no  change  greater  than  0.10  feet  in  any  single  day. 

Consider  the  Cleveland  graphs.  On  plates  7  and  8,  though  the  con- 
tinuous graph  is  smoother,  as  a  whole,  than  the  dot-and-dash  graph,  the 
contrast  is  not  great.  On  plate  9  the  small  fluctuations  in  the  continuous 
graph  for  the  periods  August  28  to  September  10  and  September  29  to  Oc- 
tober 9,  1910,  are  accompanied  by  much  larger  fluctuations  in  the  dot-and- 
dash  graph.  On  plate  10  the  same  contrast,  indicating  much  greater  accu- 
racy for  the  corrected  elevations  than  for  the  observed  elevations,  shows  for 
the  period  October  21-31,  1910. 

As  the  corrections  for  wind  effects  and  barometric  effects  at  Cleveland 
are  in  general  much  smaller  than  at  Buffalo,  it  is  to  be  expected  that  the  im- 
provement in  accuracy  produced  by  applying  the  corrections  will  be  less 
noticeable  at  Cleveland  than  at  Buffalo,  and  hence  that  the  contrast  between 
the  dot-and-dash  graph  and  the  continuous  graph  will  be  much  less  pro- 
nounced at  Cleveland  that  at  Buffalo. 

So,  too,  one  may  naturally  expect  the  contrast  between  the  two  graphs  to 
be  still  less  for  the  Lake  Erie  means  as  shown  on  the  third  pair  of  graphs  on 
plates  7  to  10  than  for  either  Buffalo  or  Cleveland.  Nevertheless,  the 
contrast  even  on  these  curves  shows  clearly  in  favor  of  the  corrected  ele- 
vations. In  this  connection,  note  especially  the  decided  smoothness  of  the 
continuous  graph  for  the  Lake  Erie  mean  for  June  1-23,  1910,  on  plate  7  as 
compared  with  the  dot-and-dash  graph,  and  the  same  contrast  on  plate  10 
for  the  stormy  period  October  11-31,  1910. 

Compare  the  continuous  graphs  on  plates  11,  12,  and  13  with  the  dot- 
and-dash  graphs.  For  each  of  the  three  separate  gages — Mackinaw, 
Harbor  Beach,  and  Milwaukee — it  is  clear  that  the  continuous  graph  is 
decidedly  smoother,  on  the  whole,  than  the  dot-and-dash  graph.  In  this 
connection,  attention  is  especially  invited  to  the  period  June  23-July  15, 
1911,  shown  on  the  lower  half  of  plate  11;  to  the  period  July  22-26,  1911, 
shown  on  plate  12;  and  to  the  period  August  28-September  19,  1911, 
shown  on  plate  13.  The  dot-and-dash  graph  for  Lake  Michigan-Huron 
mean  is  a  remarkably  smooth  curve.  It  so  closely  approaches  in  smoothness 
to  the  dot-and-two-dashes  curve  for  the  same  lake  (the  rainfall -{-inflow 
—  outflow  graph)  that  it  is  necessarily  difficult  to  tell  whether  the  con- 
tinuous graph  is  intermediate  in  smoothness.  The  graphic  method  of 
comparison  is  not  sufficiently  sensitive  to  determine  reliably  the  relative 
accuracy  in  this  case,  in  which  both  graphs,  for  the  Lake  Michigan-Huron 


108 


EFFECTS   OF   WINDS   AND    OF 


mean  observed  elevations  and  for  the  Lake  Michigan-Huron  mean  cor- 
rected elevations,  are  of  a  high  degree  of  accuracy.  The  determination  of 
relative  accuracy  must  be  made  mainly  by  other  methods  in  this  case. 
However,  an  examination  of  the  two  graphs  makes  it  clear  that  both  are  of  a 
very  high  degree  of  accuracy. 


MONTHLY  AND  SEASONAL  MEAN  ELEVATIONS. 

The  following  tables,  No.  26  for  Lake  Erie  and  No.  27  for  Lake  Michigan- 
Huron,  show  the  monthly  and  seasonal  mean  observed  elevations  and  mean 
corrected  elevations  of  the  water  surface,  first,  for  the  separate  gage  stations, 
and  in  the  last  two  columns  for  the  lake  as  derived  from  the  means  from  the 
stations  named.  These  means — monthly  and  seasonal — are  based  directly 
on  the  values  shown  in  tables  Nos.  19  to  25,  and  on  similar  tables,  which  are 
not  published,  for  the  season  of  1909  on  Lake  Erie  and  the  season  of  1910 
on  Lake  Michigan-Huron. 

TABLE  No.  26. 
[Mean  observed  and  mean  corrected  elevation  =570+  feet.] 


Buffalo  gage. 

Cleveland  gage. 

Lake  Erie. 

Year  and  month. 

Observed 

Corrected 

Observed 

Corrected 

Observed 

Corrected 

elevation. 

elevation. 

elevation. 

elevation. 

elevation. 

elevation. 

1909,  Aug  

2.68 

2.63 

2.80 

2.80 

2.75 

2.75 

1909,  Sept  

2.30 

2.25  . 

2.36 

2.36 

2.31 

2.32 

1909,  Oct  

2.05 

1.82 

1.76 

1.82 

1.90 

1.81 

1910,  June..    .. 

2.55 

2.49 

2.61 

2.59 

2.57 

2.56 

1910,  July..    .. 

2.37 

2.26 

2.40 

2.42 

2.39 

2.33 

1910,  Aug  

2.13 

2.06 

2.22 

2.28 

2.18 

2.20 

1910,  Sept  

1.83 

1.80 

2.02 

2.01 

1.91 

1.94 

1910,  Oct  

1.92 

1.70 

1.88 

1.92 

1.88 

1.84 

1909,          whole 

season,  Aug.- 

Oct 

2.34 

2.23 

2.31 

2.33 

2.32 

2.29 

1910,          whole 

season,  June- 

Oct  

2.16 

2.06 

2.22 

2.24 

2.19 

2.17 

PROBABLE  ERRORS  AND  WEIGHTS. 

The  residuals  as  tabulated  in  tables  Nos.  19  to  25,  inclusive,  are  evidently 
a  test  of  the  accuracy  of  the  observed  elevations  and  of  the  corrected  ele- 
vations corresponding  to  these  residuals.  It  is  desirable  to  study  these 
residuals  and  formulate  the  conclusions  from  them. 

If  N  independent  determinations  are  made  of  a  quantity  which  is  constant, 
and  the  residuals  are  taken  of  the  separate  determinations  from  the  mean  of 


BAROMETRIC   PRESSURES   ON   THE   GREAT   LAKES 


109 


the  N  determinations,  then  the  probable  error  of  one  determination  may  be 
computed  from  the  formula 


in  which  St>2  stands  for  the  sum  of  the  squares  of  the  residuals  in  the  group. 

TABLE  No.  27. 
[Mean  observed  and  mean  corrected  elevation =579+  feet.l 


Milwaukee  gage. 

Harbor  Beach 
gage 

Mackinaw  gage. 

Lake  Michigan- 
Huron. 

Year  and 
month.* 

Ob- 

Cor- 

Ob- 

Cor- 

Ob- 

Cor- 

Ob- 

Cor- 

served 

rected 

served 

rected 

served 

rected 

served 

rected 

eleva- 

eleva- 

eleva- 

eleva- 

eleva- 

eleva- 

eleva- 

eleva- 

tion. 

tion. 

tion. 

tion. 

tion. 

tion. 

tion. 

tion. 

1910,  June  . 

1.56 

.59 

1.54 

1.50 

1.48 

1.49 

1.52 

1.50 

1910,  July  . 

1.49 

.51 

1.42 

1.41 

1.41 

1.41 

1.44 

1.43 

1910,  Aug.  . 

1.31 

.29 

1.33 

1.33 

1.30 

1.28 

1.31 

1.30 

1910,  Sept.. 

1.29 

.28 

1.20 

1.24 

1.24 

1.22 

1.24 

1.24 

1911,  June  . 

1.05 

.06 

.98 

.98 

.98 

.98 

1.00 

1.00 

1911,  July.. 

.89 

.98 

.93 

.95 

.97 

.92 

.93 

.94 

1911,  Aug.  . 

.85 

.87 

.85 

.85 

.84 

.84 

.85 

.85 

1911,  Sept.. 

.75 

.74 

.58 

.61 

.59 

.58 

.64 

.62 

1910,  whole 

season.  .  . 

1.41 

1.42 

1.37 

1.37 

1.36 

1.35 

1.38 

1.37 

1911,  whole 

season.  .  . 

0.89 

.91 

.84 

.85 

.84 

.83 

.86 

.85 

The  probable  error  of  a  single  observed  elevation  at  each  of  the  five 
stations  was  computed  in  turn  from  this  formula  applied  to  each  group  of 
N  residuals  as  shown  in  the  tables  Nos.  19  to  23.  It  will  be  noted  that  N 
was  usually  5,  but  was  sometimes  6,  for  the  group  at  the  end  of  a  month,  and 
sometimes  less  than  5  where  observations  were  missing  or  were  rejected. 
The  mean  of  these  many  values  of  the  probable  error  of  a  single  determina- 
tion, one  for  each  group,  was  taken  as  the  probable  error  of  a  single  deter- 
mination for  that  station,  of  the  kind  under  consideration,  observed  or 
corrected. 

The  probable  errors  so  computed  are  shown  in  table  No.  28. 

When  one  considers  the  probable  errors  for  an  observed  elevation  as 
shown  for  each  of  the  five  gage  stations  in  table  No.  28,  the  question  naturally 
arises,  Why  should  the  observations  at  the  different  stations  on  any  lake  be 
given  equal  weight?  Two  considerations  led  to  the  decision  to  assign  equal 
weights  to  the  observed  elevations  in  this  investigation.  First,  that  has 
been  the  usual  practice,  so  far  as  the  investigator  knows,  in  connection  with 
other  studies  on  the  Great  Lakes.  It  did  not  seem  desirable  to  depart  from 
past  practice  except  for  clearly  good  reasons.  Second,  it  was  clearly 


110 


EFFECTS   OF   WINDS   AND    OF 


evident,  throughout  this  investigation,  that  the  observations  at  any  one  gage, 
considered  as  an  attempt  to  secure  the  mean  elevation  of  the  whole  lake 
surface,  are  subject  to  systematic  errors  which  are  not  small  in  comparison 
with  the  accidental  errors,  and  that,  therefore,  the  computed  probable 
error  of  a  single  observed  elevation,  such  a  probable  error  as  those  shown  in 
table  No.  28,  is  unreliable  as  an  indication  of  relative  accuracy  at  different 
stations.  Among  the  systematic  errors  to  which  the  observed  elevations 
are  subject  are  the  barometric  effects  and  wind  effects.  It  is  not  at  all 
certain  that  these  and  other  systematic  errors  would  be  more  thoroughly 
eliminated  from  the  mean  observed  elevations  by  using  weights  based  on 
the  probable  errors  than  by  using  equal  weights  at  the  different  stations. 
In  fact,  which  procedure  would  give  the  best  elimination  depends  largely 
upon  the  relations  to  each  other  of  the  systematic  errors  at  the  various  sta- 
tions as  well  as  upon  the  accidental  errors.  Incidentally,  it  is  interesting 
to  note  here  that  evidence  developed  very  late  in  this  investigation  as  to  the 
barometric  effects  on  Lake  Michigan-Huron  shows  clearly  that  the  alge- 
braic sum  of  the  barometric  effects  at  the  three  stations — Milwaukee,  Harbor 
Beach,  and  Mackinaw — tends  to  be  nearly  zero  on  each  day.*  Therefore, 
it  is  clear  that  equal  weights  assigned  to  the  three  observed  elevations  on 
any  day  at  these  stations  tends  to  give  a  much  better  elimination  of  the 
principal  systematic  error — that  from  barometric  effects — than  if  unequal 
weights  were  used. 

TABLE  No.  28. — Probable  error  of  a  single  daily  elevation  of  the  water  surface  computed 
from  the  residuals  shown  in  tables  19  to  £5. 


For  an 
observed  elevation. 

Fora 
corrected  elevation. 

feet. 
±  147 

feet. 
±  036 

Cleveland  gage       

±.059 

±  028 

±  054 

±  026 

Harbor  Beach  gage  

±.056 
±  039 

±.021 
±  016 

On  the  other  hand,  when  one  tests  the  corrected  elevations  in  various 
ways,  little  evidence  is  found  of  systematic  errors.  The  errors  in  these 
values  seem  to  be  mainly  of  an  accidental  character.  Hence,  for  corrected 
elevations  each  station  has  been  given  a  weight  inversely  proportional  to  the 
square  of  its  probable  error  as  shown  in  table  No.  28.  Unit  weight  was 
assigned  to  Buffalo,  where  the  probable  error  is  ±0.036  foot.  The  weights 
assigned  on  this  basis  for  other  stations  are  stated  on  pages  100,  104. 

*  Consult  table  No.  7,  page  33,  and  note  that  the  algebraic  sum  of  the  three  values  of 
Cw  for  Milwaukee,  Harbor  Beach,  and  Mackinaw,  namely,  -4.97,  +6.94,  and  —2.54,  is 
only  —0.57.  Similarly,  note  that  the  algebraic  sum  of  the  three  values  of  Cn  is  only  +3.62. 
The  smaller  these  two  sums  the  smaller  the  sum  of  the  barometric  effects  at  the  stations 
tends  to  be  on  each  day. 


BAROMETRIC   PRESSURES   ON   THE   GREAT  LAKES  111 

Among  the  values  of  the  corrected  elevations  shown  in  tables  Nos.  19  to 
23  there  is  an  occasional  one  (inclosed  in  parentheses)  which  has  been  re- 
jected by  a  definite  criterion,  as  referred  to  on  page  79,  and  therefore  has  no 
influence  upon  the  five-day  mean  or  any  of  the  other  later  means,  monthly 
or  seasonal.  That  criterion  is  of  the  character  already  described  in  general 
terms  on  page  73  in  connection  with  wind  effects  and  on  page  66  in  con- 
nection with  barometric  effects.  It  is  intended  to  identify  values  which  are 
decidedly  abnormal,  due  to  an  influence  which  extends  over  a  single  day 
only,  believed  usually  to  be  due  either  to  the  first  (and  very  large)  half 
wave  of  a  new  seiche,  started  by  a  new  powerful  wind  impulse  or  barometric 
impulse,  or  to  a  wide  departure  of  fact  from  the  assumptions  used  in  com- 
puting the  barometric  effects.  Such  wide  departures  are  believed  to  occur, 
as  a  rule,  when  a  well-developed  area  of  low  pressure  is  over  or  near  the  lake. 

The  criterion  as  applied  to  the  corrected  elevations  shown  in  tables  Nos. 
19  to  23  was  as  follows: 

A  corrected  elevation  for  a  given  day  is  to  be  rejected  whenever  it  differs 
from  that  for  the  next  preceding  or  next  following  day  by  more  than  3.5 
times  the  probable  error  of  the  change  for  one  day  as  computed  in  the  final 
least-square  solution  for  barometric  effects  at  that  station,  provided,  also, 
that  after  such  rejection  the  apparent  change  of  elevation  then  showing  for 
the  two-day  interval  covering  the  rejection  is  inside  the  3.5  limit  just  stated. 

The  limit  used  in  the  criterion  for  each  station — 3.5  times  the  probable 
error  named — was,  in  feet,  for  Buffalo  ±0.15,  for  Cleveland  ±0.14,  for 
Milwaukee  ±0.10,  for  Harbor  Beach  ±0.09,  and  for  Mackinaw  ±0.08. 

In  a  few  rare  cases  the  criterion  gave  somewhat  ambiguous  results.  In 
these  cases  the  ambiguity  was  removed  by  considering  the  residuals  from 
the  five-day  means. 

This  criterion  is  believed  to  be  reasonably  reliable  as  a  means  of  identify- 
ing abnormal  values,  and  thus  improving  the  final  accuracy  by  preventing 
any  influence  being  carried  forward  from  these  abnormal  values  into  the 
final  computed  values. 

Attention  was  called  on  page  108  to  the  fact  that  the  method  of  com- 
puting the  probable  errors  there  indicated  is  based  upon  the  assumption  that 
the  quantity  observed  is  a  constant.  Consider  this  assumption  in  connec- 
tion with  the  corrected  elevations.  The  corrected  elevation  is  supposed  to 
be  the  elevation  of  the  mean  surface  of  the  lake  in  question.  Said  elevation 
of  the  mean  surface  of  the  lake  is  certainly  not  a  constant.  It  varies  as  the 
total  water  content  of  the  lake  changes.  It  is  therefore  subject  to  con- 
tinual changes  due  to  rainfall  on  the  lake  surface,  inflow  into  the  lake  from 
the  next  lake  above  in  the  chain  of  Great  Lakes,  outflow  from  the  lake  to 
the  next  lake  below  in  the  chain,  run-off  into  the  lake  from  the  surrounding 
land,  and  evaporation  from  the  lake  surface.  Hence,  the  elevation  of  the 
mean  lake  surface  is  in  general  continually  fluctuating,  due  to  the  causes 
enumerated.  The  residuals  from  the  five-day  means  are  thereby  affected  in 
such  wise  as  to  be  increased  on  an  average  over  what  they  would  otherwise 


112  EFFECTS  OF  WINDS  AND  OF 

be.  The  computed  probable  errors  are  therefore  somewhat  too  large  to  rep- 
resent the  true  degree  of  accuracy  of  a  single  corrected  elevation. 

From  various  sources  of  information,  some  of  them  external  to  this  in- 
vestigation, the  following  estimates  of  the  fluctuations  of  mean  lake  surface 
of  Lake  Erie  have  been  made.  The  estimates  are  approximately  applicable 
to  Lake  Michigan-Huron,  with  the  one  exception  that  the  net  outflow  from 
Lake  Michigan-Huron  is  smaller,  in  the  units  here  used,  than  from  Lake 
Erie  and  less  variable. 

It  is  estimated  for  Lake  Erie  that  during  the  months  June  to  October  of 
each  year  the  net  outflow,  namely,  the  outflow  through  the  Niagara  River 
minus  the  inflow  through  the  Detroit  River,  corresponds  to  a  fall  in  the  mean 
lake  surface  varying  from  +0.016  foot  on  some  days  to  —0.002  foot  at  the 
other  extreme  for  some  days.  Similarly,  it  is  estimated  that  the  rainfall  on 
the  lake  surface  causes  a  rise  in  the  mean  lake  surface  varying  from  zero  on 
days  of  no  rain  to  +0.085  on  rare  days  of  very  heavy  rainfall;  that  the  run- 
off into  the  lake  from  the  surrounding  land  corresponds  to  a  rise  varying 
from  +0.004  foot  on  some  days  to  +0.040  on  other  days;  and  that  the 
evaporation  corresponds  to  a  fall  in  the  mean  lake  surface  of  from  0.000 
foot  on  some  days  to  +0.021  foot  on  days  of  extremely  rapid  evapora- 
tion. 

For  the  season  of  1910,  the  five  months  June  to  October,  inclusive,  the 
algebraic  sum  of  these  four  influences  was  a  fall  of  about  0.004  foot  per  day 
for  the  whole  season. 

An  inspection  of  the  monthly  means  of  corrected  elevations  as  shown  in 
table  No.  26  makes  it  clear  that  in  general  during  the  season  June  to  October 
the  mean  lake  surface  is  falling  at  a  mean  rate  usually  between  0.00  and 
0.02  foot  per  day. 

It  is  clear  from  the  three  paragraphs  next  preceding  this  that  the  actual 
variation  of  the  mean  elevation  of  the  whole  surface  of  any  one  of  the  Great 
Lakes  is,  as  a  rule,  as  much  as  0.01  foot  in  two  days,  that  it  is  frequently 
more  than  0.02  foot  in  24  hours,  and  that  on  rare  occasions  it  may  exceed 
0.08  in  that  period. 

This  estimated  rate  of  variation  was  taken  into  account  in  fixing  the  size 
of  the  group  to  be  used  in  taking  each  mean  in  such  tables  as  Nos.  19  to  24. 
If  the  group  were  taken  very  small,  as,  for  example,  two  or  three  days  only, 
the  mean  would  be  unstable,  and  the  two  or  three  residuals  would  give  a  very 
poor  computed  value  for  the  probable  error.  On  the  other  hand,  if  the  group 
were  made  large,  to  include,  say,  eleven  values,  then  the  extreme  residuals  in 
each  group  would  depend  very  largely  upon  the  actual  variations  in  the  mean 
lake  surface  commented  upon  in  the  four  preceding  paragraphs,  and  so  would 
not  be  of  value  as  indicators  of  the  accuracy  of  the  corrected  elevation  for 
each  day.  In  an  eleven-day  group,  the  extreme  residual  at  each  end  of  the 
group  would  include  the  variation  in  elevation  of  the  mean  lake  surface  for 
five  days,  the  interval  between  that  residual  and  the  middle  date  of  the  group. 
For  the  mean  for  the  group  most  nearly  represents  the  middle  date  in 


BAROMETRIC  PRESSURES  ON  THE  GREAT  LAKES     113 

general.  The  final  decision  was  to  use  groups  of  five  days  each  as  being  the 
best  available  compromise  between  the  two  difficulties.  The  end  residuals 
of  each  five-day  group  include,  in  general,  the  actual  fluctuation  in  mean 
lake  elevation  for  two  days  only. 

With  the  five-day  groups,  and  on  the  basis  of  the  estimate  stated  in  the 
second  paragraph  before  this,  it  appears  that  of  the  end  residuals  of  each 
group  0.01  foot  is  usually  due  to  actual  fluctuation  in  the  elevation  of  the 
mean  surface  of  the  lake.  In  a  few  cases  as  much  as  0.04  foot  or  more  of 
such  end  residuals  may  be  due  to  such  fluctuation. 

From  the  considerations  which  have  been  indicated  briefly  above,  the 
writer  estimates  that  the  probable  errors  shown  in  the  last  column  of  table 
No.  28  are  appreciably  too  large  to  represent  the  true  degree  of  accuracy  of 
the  corrected  elevations.  It  is  possible  that  the  probable  error  of  a  cor- 
rected elevation  for  one  day  for  the  mean  surface  of  the  whole  of  Lake 
Michigan-Huron,  as  derived  from  observations  at  the  Mackinaw  gage,  is  as 
small  as  ±0.010  foot,  instead  of  the  value  ±0.016  foot,  as  shown  in  table 
No.  28.  Much  better  determinations  of  these  probable  errors  will  become 
available  when  the  investigation  of  evaporation  is  made  from  this  data. 
Until  that  time,  one  must  be  content  with  the  approximate  estimate  of  this 
paragraph  and  the  positive  knowledge  that  the  values  in  the  last  column  of 
table  No.  28  are  too  large  to  represent  the  truth. 


TIDES. 

The  question  may  properly  be  raised,  Do  the  true  tides  produced  in  the 
Great  Lakes  by  the  moon  and  sun  have  any  appreciable  effect  upon  this 
investigation?  No  account  has  been  taken  of  such  tides  anywhere  in  this 
investigation.  The  true  tide  at  Chicago  and  at  Milwaukee  produces  a  total 
range  of  oscillation  of  0.14  foot  or  less.  (See  Coast  and  Geodetic  Survey 
Report  for  1907,  pages  483-486,  in  the  Manual  of  Tides,  by  R.  A.  Harris.) 
This  tidal  oscillation  is  made  up  of  several  components,  each  approximately 
a  sine  wave,  and  each  with  a  period  which  is  either  approximately  12  solar 
hours  or  12  lunar  hours,  or  24  solar  hours  or  24  lunar  hours.  Such  an  oscil- 
lation has  practically  no  effect  on  the  mean  for  each  day.  Certainly  it  does 
not  affect  it  by  as  much  as  0.01  foot  in  any  case.  At  other  stations  on  the 
Great  Lakes  the  tide  is,  as  a  rule,  smaller  than  at  Chicago  and  Milwaukee, 
and  is  nowhere  much  larger.  The  true  tides  produce  an  appreciable  effect 
on  the  hourly  elevations  of  the  water  surface  such  as  were  used  in  determin- 
ing the  wind  effects,  reaching  a  maximum  probably  not  greater  than  0.10 
foot  for  any  hour.  But  the  determination  of  wind  effects  is  essentially  based 
on  the  rate  of  change  of  elevation  from  hour  to  hour.  That  rate  of  change 
is  so  slightly  affected  by  the  true  tides  and  enters  so  nearly  as  an  accidental 
error  positive  and  negative  with  almost  exactly  the  same  frequency,  that 
the  final  conclusion  as  to  wind  effects  is  certainly  not  affected  appreciably 
by  the  tides. 


114  EFFECTS   OF   WINDS   AND    OF 

SEICHES. 

A  seiche  is  an  oscillation  in  the  waters  of  the  lake  under  the  influence  of 
inertia.  It  is  a  free  oscillation  as  distinguished  from  a  forced  oscillation. 
It  is  a  wave  motion  involving  both  horizontal  transfer  of  water  back  and 
forth  and  a  vertical  oscillation  of  the  water  surface. 

The  seiches  of  Lake  Erie,  and  to  a  lesser  extent  of  Lake  Michigan-Huron, 
were  studied  in  this  investigation  because  it  became  evident  that  they  are 
one  of  the  principal  sources  of  error.  It  seemed  probable  that  the  more 
clearly  the  seiches  were  understood  the  more  effectually  could  one  guard 
against  the  occasional  abnormally  large  errors  arising  from  this  source. 

The  seiches  were  studied  principally  in  three  ways : 

They  were  studied  by  graphs  such  as  are  shown  on  plates  7  to  16.  These 
graphs  aided  especially  in  finding  the  relation  of  a  given  seiche  to  the  impulse 
which  started  it  and  in  indicating  how  rapidly  the  oscillation  died  out  and 
under  what  conditions. 

The  seiches  were  studied  by  statistical  methods,  mainly  for  the  purpose  of 
determining  the  prevailing  periods  of  oscillation. 

The  seiches  were  studied  by  computing  the  theoretical  period  of  oscil- 
lation of  the  lake,  or  of  certain  parts  of  it,  from  the  known  dimensions  of  the 
lake,  horizontal  distances  and  depths.  This  served  to  establish  the  probable 
manner  of  oscillation  by  identifying  an  already  determined  period  as  per- 
taining to  a  particular  manner  of  oscillation. 

SEICHES  AT  BUFFALO. 

At  Buffalo  22  days  of  hourly  elevations  of  the  water  surface  observed 
at  the  gage  were  studied  in  detail  for  seiches.  These  particular  22  days  had 
been  especially  selected  to  include  periods  of  unusually  large  barometric 
effects  and  wind  effects,  especially  the  latter.  It  was  to  be  presumed,  there- 
fore, that  these  days  included  the  impulses  which  started  many  new  seiches 
of  unusual  size. 

From  the  observed  hourly  elevations  corrected  for  wind  effects  and  baro- 
metric effects,  according  to  best  evidence  then  available,  and  the  correspond- 
ing graphs  which  had  been  drawn  for  all  22  days,  a  table  was  made  showing 
every  maximum  point  and  every  minimum  point.  The  intervals  of  time 
from  each  maximum  to  the  next  following  maximum  and  from  each  mini- 
mum to  the  next  following  minimum  were  also  tabulated.  Each  of  these 
intervals  is  the  period  of  one  of  the  complete  oscillations  which  actually  took 
place  about  the  equilibrium  points  instantaneously  fixed  by  the  wind  and 
barometric  effects.  Each  of  these  oscillations  is  probably  in  general  a 
composite  of  two  or  more  seiches  of  different  periods  and  of  the  forced 
oscillation  produced  by  the  changing  winds  and  changing  barometric  con- 
ditions during  that  time  interval.  The  immediate  problem  which  confronted 
the  investigator  was  to  detect  from  a  study  of  22  days  of  these  composites 
the  seiche  periods  which  prevail  at  Buffalo. 


BAROMETRIC  PRESSURES  ON  THE  GREAT  LAKES 


115 


A  frequency  table  was  constructed  from  the  observed  periods  referred  to 
in  the  preceding  paragraph.     It  is  shown  below  as  table  No.  29. 

TABLE  No.  29. — Frequency  distribution  of  observed  periods  of  oscillation  at  Buffalo. 


First. 

Second. 

Third. 

Length  of 
observed 
periods 
in  hours. 

Number  of 
such  periods 
observed. 

Length  of 
observed 
periods 
in  hours. 

Number  of 
such  periods 
observed. 

Length  of 
observed 
periods 
in  hours. 

Number  of 
such  periods 
observed. 

2 

24 

6.5 

3 

11 

1 

2.5 

4 

7 

6 

12 

3 

3 

44 

7.5 

1 

12.5 

1 

3.5 

3 

8 

4 

13 

2 

4 

53 

8.5 

1 

14 

1 

4.5 

2 

9 

7 

14.5 

1 

5 

17 

10 

3 

15.5 

1 

6 

14 

161  periods  in  first  group,  mean  period  =  3. 7  hours. 
10  periods  in  third  group,  mean  period  =  13.0  hours. 
Total  number  of  periods  196. 

From  a  study  of  this  table,  combined  with  a  study  of  the  graphs,  it  ap- 
peared that  probably  the  161  periods  included  in  the  first  part  of  the  table 
were  all  disturbed  cases  of  one  prevailing  seiche  period  between  3  and  4 
hours  in  length.  Accordingly,  the  mean  of  these  161  peroids,  namely,  3.7 
hours,  was  taken  as  a  best  first  approximation  to  the  period  of  a  prevailing 
seiche  at  Buffalo. 

The  periods  involving  half  hours,  such  as  2.5,  3.5,  4.5,  etc.,  arose  from 
the  fact  that  where  two  successive  hourly  ordinates  at  a  maximum  or  a 
minimum  on  the  graph  were  equal,  the  time  of  said  maximum  or  minimum 
was  identified  as  being  at  the  half  hour  midway  between  the  two  hours. 
Naturally,  there  were  relatively  few  of  these  cases  as  compared  with  those 
in  which  the  observed  period  was  an  integral  number  of  hours.  The  fre- 
quency-distribution table,  No.  29,  should  therefore  be  scanned  as  if  these 
periods  involving  half  hours  were  distributed  equally  to  the  adjacent  periods 
which  are  in  integral  hours.  The  table  so  scanned  shows  a  maximum  fre- 
quency at  4  hours. 

A  study  of  the  graphs  showed  several  cases  in  which  an  apparent  seiche 
with  a  period  of  about  13  hours  stood  out  clearly  with  relatively  little  com- 
plication by  anything  else.  With  this  as  a  clue,  it  was  decided  that  possibly 
the  10  periods  included  in  the  third  part  of  the  table  were  all  disturbed 
cases  of  a  seiche  with  a  period  of  about  13  hours.  Accordingly,  the  mean 
of  these  10  periods,  namely,  13.0  hours,  was  taken  as  a  best  first  approxi- 
mation to  the  period  of  a  prevailing  seiche  at  Buffalo. 

Next,  the  25  cases  in  the  middle  of  the  frequency-distribution  table,  No. 


116  EFFECTS  OF  WINDS  AND  OF 

29,  were  inspected  in  detail  on  the  graphs.  This  inspection  showed  evidence 
in  16  cases  out  of  the  25  that  a  short-period  oscillation  (3.7  hours)  and  a 
long-period  oscillation  (13.0  hours)  were  apparently  both  in  progress  at  the 
same  time.  In  five  other  cases  out  of  the  25  the  inspection  showed  evidence 
that  an  oscillation  with  a  period  of  about  3.7  hours  was  apparently  in  progress 
complicated  by  other  things,  but  no  13-hour  period  was  evident. 

At  this  point  in  the  investigation  it  appeared  to  be  probable  that  the  13- 
hour  seiche  was  almost  or  quite  continuously  present  at  Buffalo,  usually 
more  or  less  masked  by  other  seiches  and  by  forced  oscillations.  Accord- 
ingly, the  principal  maxima  and  minima  were  selected  from  among  the 
196  tabulated  in  table  No.  29  to  cover  the  whole  22  days  and  to  correspond 
to  the  supposition  of  the  preceding  sentence.  From  these  selected  principal 
maxima  a  second  determination  of  the  long  period  became  available,  and, 
similarly,  from  the  selected  principal  minima  a  third  determination. 

The  three  determinations  of  the  period  of  the  long  Buffalo  seiche  as  thus 
determined  were:  Period 

equals 

From  frequency-distribution  table,  No.  29,  mean  of  10  cases 13.0  hrs. 

From  principal  maxima  throughout  the  22  days,  28  cases 13.4  hrs. 

From  principal  minima  throughout  the  22  days,  31  cases 12 . 8  hrs. 


Mean 13.1  hrs. 

The  mean  13.1  hours  was  adopted  as  the  most  probable  value  of  the 
period  of  the  long  seiche,  and  its  uncertainty  was  estimated  at  not  much 
more  than  0.1  hour. 

Again,  the  graphs  were  carefully  studied  in  detail  for  evidence  as  to 
apparent  damping  of  the  13.1  hour  seiche  and  for  evidence  as  to  the  impulses 
which  initiate  it.  Three  conclusions  were  reached : 

(1)  Each  new  large  13.1  hour  seiche  is  preceded  by  a  large  discernible 
impulse  from  the  wind  or  from  barometric  pressure.  (2)  Large  13.1  hour 
seiches  are  subject  to  rapid  apparent  damping  even  though  some  new  impulse 
occurred.  (3)  A  wind  impulse,  if  sufficient  to  change  the  elevation  of  the 
water  surface  corresponding  to  equilibrium  by  more  than  0.20  foot,  ordina- 
rily either  started  a  new  13.1  hour  seiche  or  clearly  distorted  such  a  seiche 
which  was  already  in  progress. 

SEICHES  AT  CLEVELAND. 

At  Cleveland  22  days  of  hourly  elevations  of  the  water  surface  observed 
at  the  gage  were  studied  in  detail  for  seiches.  These  particular  22  days  had 
been  especially  selected  to  include  periods  of  unusually  large  barometric 
effects  and  wind  effects,  especially  the  latter.  It  was  to  be  presumed,  there- 
fore, that  these  days  included  the  impulses  which  started  many  new  seiches 
of  unusual  size. 

The  study  at  Cleveland  was  somewhat  similar  to  that  indicated  above  as 
having  been  made  at  Buffalo.  The  frequency-distribution  table  for  Cleve- 


BAROMETRIC  PRESSURES  ON  THE  GREAT  LAKES     117 

land,  analogous  to  table  No.  29,  showed  clearly  a  short-period  seiche,  but 
furnished  no  clear  evidence  of  a  long-period  seiche.  There  were  252  periods 
in  the  table.  Of  these  252  periods,  165  were  not  less  than  2  hours  nor  more 
than  3.5  hours.  The  mean  of  these  165  periods  was  2.6  hours,  which  is 
believed  to  be  the  prevailing  seiche  at  Cleveland. 

A  study  of  the  Cleveland  graphs  indicated  that  a  long-period  seiche  might 
be  in  existence  there  much  of  the  time,  and  yet  little  or  no  evidence  of 
it  would  appear  in  a  frequency-distribution  table  similar  to  table  No.  29, 
because  the  short-period  seiche  (2.6  hours)  was  rather  persistent  and  be- 
cause its  shortness  of  period  tended  to  produce  well-defined  maxima  and 
minima,  which  would  tend  to  conceal  in  such  a  table  any  long-period  seiche. 
Accordingly,  three  determinations  of  a  long  seiche  period  at  Cleveland  were 
made  by  the  method  of  selected  principal  maxima  and  minima.  The 
selection  was  made  by  somewhat  arbitrary  rules  based  on  the  general  prop- 
osition that  the  purpose  of  the  selection  was  to  remove  the  mask  made  up 
of  short-period  oscillations,  mainly  2.6  hour  seiches.  The  first  determination 
was  based  on  one  selection  of  a  moderate  number  of  principal  maxima  and 
minima  and  a  frequency-distribution  table.  The  second  and  third  determi- 
nations were  based  on  a  second  selection  of  principal  maxima  and  minima. 
The  three  determinations  involved,  respectively,  29,  31,  and  30  identified 
periods.  The  three  mean  periods  from  the  three  determinations  were  in 
order  13.9  hours,  12.5  hours,  and  12.9  hours.  The  mean  of  these  is  13.1  hours, 
which  is  believed  to  be  the  period  of  the  prevailing  long  seiche  at  Cleveland. 

The  determination  of  this  period,  13.1  hours,  at  Cleveland,  as  indicated 
above,  is  weak. 

But  there  are  three  supporting  lines  of  evidence  indicated  below,  which 
greatly  strengthen  the  evidence  indicated  above  in  favor  of  a  13.1  period  at 
Cleveland. 

First,  it  is  to  be  noted  that  the  mean,  13.1  hours,  of  three  widely  separated 
values,  13.9,  12.5,  and  12.9,  agrees  exactly  with  the  13.1  hour  period  de- 
termined independently  at  Buffalo  on  the  same  lake. 

Secondly,  there  were  6  days  of  the  22  examined  at  Buffalo  and  Cleve- 
land which  were  common  to  these  two  stations.  On  these  6  days  the  maxima 
at  Cleveland  of  the  supposed  13.1  hour  seiche  coincided  in  time,  within  the 
errors  of  identification  at  the  two  stations,  with  the  minima  at  Buffalo  of  the 
supposed  13.1  hour  seiche  there.  So,  also,  the  minima  at  Cleveland  co- 
incided closely  with  the  maxima  at  Buffalo. 

Thirdly,  the  theoretical  period  of  oscillation  of  Lake  Erie  lengthwise  was 
computed  and  found  to  be  13.1  hours.  This,  combined  with  the  second  line 
of  evidence,  clearly  identifies  the  Cleveland  and  Buffalo  13.1  hour  seiches 
as  one  and  the  same  thing,  as  the  theory  shows  that  in  that  case  the  relation 
of  maxima  and  minima  at  the  two  stations  should  be  that  which  was  actually 
found. 

The  theoretical  period  of  oscillation  of  Lake  Erie  lengthwise  was  found  by 
use  of  the  table  on  page  618  of  the  Coast  and  Geodetic  Survey  Report  for 


118  EFFECTS  OF  WINDS  AND  OF 

1897.  This  table  is  based  on  the  well-known  laws  of  the  rate  of  propagation 
of  a  tidal  wave,  or,  in  general,  of  a  wave  in  water  of  which  the  depth  is  small 
in  comparison  with  the  length  of  the  wave.  In  that  case,  as  shown  in  the 
table,  the  rate  of  propagation  of  the  wave  is  purely  a  function  of  the  depth 
of  the  water.  For  a  statement  of  the  theory  involved,  especially  as  applied 
to  seiches,  consult  page  348,  in  the  volume  referred  to,  in  the  Manual  of 
Tides,  part  7,  by  R.  A.  Harris. 

It  was  known  to  the  investigator,  also  on  the  basis  of  Dr.  Harris's  tidal 
theory,  that  in  a  lengthwise  oscillation  of  the  lake  as  a  whole  the  end  limits 
between  which  the  oscillation  primarily  takes  place  would  probably  lie  not 
at  the  very  ends  of  the  lake,  but  at  the  point  near  each  end  at  which  the 
depths  begin  to  change  rapidly  as  the  shore  is  approached. 

The  mean  depth  at  each  part  of  the  lake  to  be  used  in  computing  the  rate 
of  propagation  of  the  wave  was  easily  derivable  from  the  estimates  of  depths 
already  made  in  connection  with  the  computation  of  Sx  for  Lake  Erie,  as 
described  on  pages  44^48. 

It  was  found  that  the  theoretical  period  of  oscillation  of  Lake  Erie  length- 
wise is  13.1  hours,  provided  the  two  end  limits  of  the  primary  oscillation  are 
at  the  west  end  at  a  meridian  about  7,000  feet  east  of  Cedar  Point  (near 
Sandusky),  and  at  the  east  end  about  49,000  feet  to  the  westward  of  the 
extreme  eastern  shore,  in  60  feet  of  water,  in  the  meridian  which  is  about 
midway  between  Sturgeon  Point  and  Windmill  Point.  At  the  eastern 
limit  named  the  water  begins  to  shoal  rapidly.  The  western  limit  named  is 
at  the  entrance  of  the  shallow  western  extension  of  the  lake,  in  which  the 
depths  are  everywhere  6  fathoms  or  less.  These  eastern  and  western  limits 
are  shown  by  three  stars  on  plate  2.  In  the  main  portion  of  Lake  Erie 
(see  plate  2)  the  depths  are  uniform  over  the  greater  portion  at  12  to  14 
fathoms.  There  is  a  relatively  small  portion  in  the  eastern  quarter  of  the 
lake  in  which  the  depths  are  from  20  to  34  fathoms. 

The  evidence  is  conclusive  that  the  prevailing  long-period  seiche  at  both 
Buffalo  and  Cleveland  is  a  lengthwise  oscillation  of  Lake  Erie  between  the 
limits  stated  in  the  preceding  paragraph.  This  seiche  is  of  the  type  which 
might  be  called  a  wash-basin  oscillation,  in  which  there  is  a  nodal  line  which 
changes  but  little,  if  any,  in  elevation.  The  water  surface  falls  on  one  side 
of  the  nodal  line  simultaneously  with  the  rise  on  the  other  side. 

By  a  study  similar  to  that  indicated  above  in  connection  with  the  13.1 
hour  seiche,  it  appears  to  be  probable  that  the  primary  oscillation  of  the 
3.7  hour  seiche  observed  at  Buffalo  is  a  lengthwise  oscillation  in  that  deep 
part  of  Lake  Erie  already  referred  to  as  containing  depths  of  20  to  34 
fathoms,  in  sharp  contrast  with  the  greater  part  of  the  lake,  which  has  a  very- 
flat  bottom  in  depths  of  12  to  14  fathoms.  The  computation  indicated  the 
eastern  limit  of  the  3.7  hour  oscillation  to  be  at  the  same  point  (marked  by 
three  stars  on  plate  2)  as  the  eastern  limit  of  the  13.1  hour  oscillation.  The 
western  limit  was  indicated  by  the  computation  to  be  at  the  locality  marked 
by  two  stars  on  plate  2,  in  the  most  constricted  part,  as  to  width,  of 


BAROMETRIC   PRESSURES   ON   THE   GREAT  LAKES  119 

Lake  Erie  and  where  the  depth  is  decreasing  very  rapidly  from  east  to 
west. 

Similarly,  from  a  study  of  the  depths,  the  corresponding  rates  of  propa- 
gation, and  the  probable  points  of  reflection,  it  appears  that  the  2.6  hour 
seiche  observed  at  Cleveland  is  probably  a  crosswise  oscillation  back  and 
forth  between  the  10-fathom  curve  off  Cleveland  and  the  10-fathom  curve 
on  the  opposite  (Canadian)  shore  of  the  lake,  about  midway  between  Point 
Pelee  and  Point  aux  Pins. 

Though  the  hourly  observations  of  elevation  of  water  surface  at  Mil- 
waukee, Harbor  Beach,  and  Mackinaw  show  that  seiches  of  moderate  range 
are  prevalent  at  each  of  these  stations,  it  was  not  feasible  to  make  a  careful 
study  of  these  within  the  limits  of  this  investigation,  with  the  one  exception 
of  the  study  made  for  Mackinaw,  as  indicated  below. 

SEICHES  AT  STRAIT  OF  MACKINAC. 

The  Strait  of  Mackinac  constitutes  a  connection  between  Lake  Michigan 
and  Lake  Huron.  The  Strait  for  a  length  of  20,000  feet  at  its  narrowest 
part  has  a  width  of  about  20,000  feet  and  a  mean  depth  of  about  68  feet.  It 
appeared  to  the  investigator  that  there  was  probably  a  peculiar  seiche  hav- 
ing its  nodal  line  at  about  the  middle  point  of  the  length  of  the  narrowest 
part  of  the  Strait,  and  that  the  oscillation  would  be  such  that  the  whole 
surface  of  Lake  Michigan  would  rise  while  the  whole  surface  of  Lake 
Huron  went  down,  and  vice  versa,  as  the  current  ran  alternately  westward 
and  eastward  through  the  Strait  under  the  influence  of  inertia,  gravity, 
and  friction,  after  such  an  oscillation  or  seiche  had  once  been  started. 
The  investigator  did  not  know  of  any  adequate  treatment  of  the  problem  of 
determining  the  theoretical  period  of  such  a  seiche.  Therefore,  he  started 
from  the  known  elements  of  the  problem:  the  areas  of  the  two  lakes;  the 
Chezy  formulae  for  the  relations  between  the  velocity  of  a  steady  current 
flowing  in  a  channel,  the  slope  of  the  water  surface,  and  the  dimensions  of 
the  channel;  an  assumed  coefficient  of  roughness  for  use  in  that  formula; 
and  the  known  relations  between  mass,  force,  and  acceleration.  By  a  step- 
by-step  process  he  computed  the  period  of  oscillation  by  main  strength  and 
computed  the  probable  rate  of  damping.  The  conclusions  reached  were 
(1)  that  the  period  is  about  7  hours  for  a  complete  oscillation,  and  (2)  that 
the  damping  is  at  least  sufficient  to  reduce  the  amplitude  of  the  wave  by  one- 
sixth  part  in  each  successive  half  wave. 

After  the  computation  was  complete,  an  inspection  was  made  of  the 
graph  of  hourly  elevations  of  the  water  surface  on  42  days  at  the  Mackinaw 
gage,  which  is  at  the  eastern  end  of  the  Strait.  The  inspection  was  not  de- 
tailed or  complete,  as  the  available  time  was  short.  Two  cases  were  found 
on  which  a  7-hour  seiche  was  apparently  started  at  Mackinaw  by  an  im- 
pulse due  to  barometric  pressures,  which  was  peculiarly  well  adapted  to 
start  such  a  seiche.  It  was  also  noted  that  on  May  18-23,  1911,  there  was 


120  EFFECTS  OF  WINDS  AND  OF 

a  continuous  and  unusually  large  oscillation  at  the  Mackinaw  gage,  of  which 
the  apparent  period  between  successive  maxima  or  minima  was  either 
6,  7,  or  8  hours  in  more  than  one-third  of  the  cases.  The  mean  period  of 
oscillation  for  these  6  days  was  found  to  be  6.8  hours. 

The  computation  of  the  theoretical  period  of  oscillation  through  the 
Strait  of  Mackinac  needs  confirmation,  and  especially  needs  a  careful  study 
of  the  possible  errors  due  to  certain  necessary  assumptions  made  in  the 
computations.  So,  too,  the  evidence  from  the  necessarily  cursory  examina- 
tion of  the  graph  of  hourly  elevations  at  Mackinaw  is  weak.  Nevertheless, 
the  investigator  reached  the  conclusion,  subject  to  the  reservations  implied 
in  the  preceding  two  sentences,  that  there  is  probably  a  seiche  through  the 
Strait  of  Mackinac  of  the  character  indicated,  with  a  nodal  line  near  the 
mid-length  of  the  narrowest  part  of  the  Strait,  with  a  period  of  about  7  hours. 

It  should  be  noted,  in  closing  this  general  statement  in  regard  to  seiches, 
that  as  this  investigation  was  based  on  observed  hourly  elevations  of  the 
water  surface  it  is  incapable  of  detecting  seiches  having  periods  less  than 
1  hour. 

The  present  investigator  feels  that  there  is  a  very  interesting  field  open 
for  further  study  of  the  seiches  in  the  Great  Lakes,  and  that  rapid  progress 
is  possible  in  this  field,  which  will  lead  to  a  much  better  knowledge  than  is 
now  available  in  regard  to  seiches  and  tidal  oscillations.  But  he  finds,  with 
regret,  that  he  must  turn  aside,  after  his  little  excursion  into  the  field,  to 
carry  forward  other  lines  of  investigation  and  of  work  to  which  he  has 
already  committed  himself. 

EXAMPLES  OF   SEICHES. 

Plates  14,  15,  and  16  show  some  examples  of  seiches.  All  of  these  plates 
are  drawn  to  the  same  scale.  On  each  of  them  four  graphs  plotted  from 
hourly  ordinates  are  shown. 

The  observed  hourly  elevations  of  the  water  surface  are  shown  by  the  dot- 
and-dash  graph.  The  elevation  of  the  water  surface  at  each  hour  referred 
to  mean  sea-level  may  be  read  in  feet  from  the  scale  of  numbers  at  the  right 
margin  of  the  plate. 

The  barometric  effect  at  each  hour  is  shown  by  the  dotted  graph.  Its 
amount  in  feet  may  be  read  from  the  scale  of  numbers  at  the  left  margin  of 
the  plate. 

The  wind  effect  at  each  hour  is  shown  by  the  dashed  graph.  Its  amount 
in  feet  may  be  read  from  the  scale  of  numbers  at  the  left  margin  of  the  plate. 

The  elevation  of  the  water  surface  at  each  hour  after  correction  for  baro- 
metric effect  and  for  wind  effect  is  shown  by  the  continuous  graph.  The 
corrected  elevations  referred  to  mean  sea-level  may  be  read  from  the  scale 
of  numbers  at  the  right  margin  of  the  plate.  Each  ordinate  on  this  graph 
is  that  of  the  first  graph  (observed  elevations)  diminished  algebraically  by 
the  ordinates  of  the  second  (barometric)  and  third  (wind)  graphs.  This 


BAROMETRIC   PRESSURES   ON   THE    GREAT  LAKES  121 

fourth  (continuous)  graph  shows  the  oscillations  of  the  water  surface,  under 
the  influence  of  inertia,  about  the  instantaneous  positions  of  equilibrium  of 
the  water  surface  fixed  by  the  barometric  gradients  and  the  winds. 

Consider  the  information  shown  on  plate  14,  pertaining  to  Buffalo : 

Note  that  the  barometric  effect,  as  shown  by  the  dotted  graph,  was 
+0.22  foot  at  1  a.m.  on  August  4,  1910,  that  it  increased  steadily  to  +0.46 
foot  at  8  a.m.,  then  began  to  decrease  at  once,  and  decreased  steadily  to 
+0.27  foot  at  1  p.m.  Thereafter  the  barometric  effect  fluctuated  but  little 
and  slowly  until  1  p.m.  on  August  6.  From  1  p.m.  on  August  6  to  11  a.m. 
on  August  7  the  barometric  effect  decreased  slowly  and  steadily  from  +0.28 
foot  to  —0.11  foot.  Thereafter  the  barometric  effect  remained  nearly 
constant. 

Note  that  during  these  4  days  the  wind  effect,  as  shown  by  the  dashed 
graph,  was  very  small  until  5  a.m.  on  August  4,  increased  rapidly  after  that 
time  to  +0.32  foot  at  noon  on  August  4,  then  decreased  rapidly  to  +0.01 
foot  at  9  p.m.  on  August  4,  and  thereafter  increased  slowly  to  +0.08  at  9 
a.m.  on  August  5.  From  9  a.m.  on  August  5  the  wind  effect  increased 
rapidly  to  +0.65  foot  at  5  p.m.,  and  then  decreased  still  more  rapidly  to 
+0.03  foot  at  11  p.m.  on  August  5.  Thereafter,  during  August  6  and 
August  7,  the  wind  effect  fluctuated  but  little.  It  was  not  more  than  0.01 
foot  at  any  time  between  6  p.m.  on  August  6  and  the  end  of  August  7. 

From  the  graph  of  observed  elevations,  note  that  the  water  responded  to 
the  barometric  and  wind  impluses  in  the  forenoon  of  August  4  and  to  the 
reversed  wind  impulse  of  that  afternoon.  Note  that  the  continuous  graph 
of  corrected  elevations  is  very  irregular  after  these  impulses,  and  that  during 
the  period  9  p.m.  on  August  4  to  9  a.m.  on  August  5,  when  the  water  was 
comparatively  free  from  new  impulses,  there  is  some  indication  of  a  seiche 
with  a  period  between  3  and  4  hours  and  a  range  of  more  than  0.2  foot. 

Note  that  the  water  surface  responded  to  the  sudden  large  upward  wind 
impulse  which  occurred  between  9  a.m.  and  5  p.m.  of  August  5  and  to  the 
still  larger  and  more  sudden  downward  wind  impulse  between  5  p.m.  and 
11  p.m.  on  August  5.  Note  that  thereafter,  during  August  6  and  7,  with 
no  large  new  impluses,  the  continuous  graph  (corrected  elevations)  shows 
clearly  a  long-period  seiche  in  progress.  The  period  of  this  seiche  is  ap- 
parently slightly  in  excess  of  13  hours,  as  identified  from  the  record  of  these 
2  days.  It  is  believed  to  be  the  13.1  hour  seiche  for  which  the  evidence  is 
summarized  on  pages  114-116.  Note  that  the  total  range  of  this  seiche  on 
August  6  is  0.7  or  0.8  foot  and  on  August  7  is  about  0.5  foot.  Note  its 
comparatively  regular  form  on  August  7. 

Examine  plate  15,  showing  the  four  graphs  for  Buffalo  on  October  21-22, 
1909.  Note  that  the  water  surface,  as  shown  by  the  observed  elevations 
(the  dot-and-dash  graph),  responded  to  the  large  and  gradual  barometric 
impulse.  The  most  striking  thing  shown  on  this  plate  is  the  short-period 
seiche  which  was  evidently  in  progress  throughout  the  2  days,  continually 
being  modified  and  distorted  but  persisting.  On  the  continuous  graph 


122  EFFECTS  OF  WINDS  AND  OF 

(corrected  elevations)  there  are  maxima  on  October  21  at  4  a.m.,  7  a.m., 
9  a.m.,  noon,  6  p.m.,  8  p.m.,  and  midnight,  and  on  October  22  at  3  a.m., 
7  a.m.,  11  a.m.,  2  p.m.,  7  p.m.,  and  11  p.m.  These  apparently  define  12 
complete  waves  of  the  3.7  hour  seiche  referred  to  on  page  115.  The  different 
values  of  the  period,  from  the  12  separate  intervals  between  the  maxima 
noted,  varied  from  2  to  6  hours,  with  a  mean  for  the  12  waves  of  3.6  hours. 
Similarly,  if  one  judges  by  the  minima  on  these  2  days,  there  were  12  com- 
plete waves  between  3  a.m.  of  October  21  and  10  p.m.  of  October  22, 
no  single  wave  being  shorter  than  2  hours  and  none  longer  than  5  hours. 
The  mean  period  from  these  minima  is  one-twelfth  of  43  hours,  or  3.6 
hours. 

Examine  plate  16,  which  shows  the  four  graphs  for  Buffalo  on  October 
26-27,  1910,  and  also  in  the  upper  left  quarter  of  the  plate  the  four  graphs 
for  Cleveland  on  October  27. 

On  October  26,  1910,  at  Buffalo  a  long-period  seiche,  apparently  the  13.1 
hour  seiche,  was  in  progress,  with  a  range  of  0.5  to  0.8  foot,  and  the  baromet- 
ric effects  and  wind  effects  were  not  large  nor  changing  very  rapidly. 

Between  11  p.m.  on  October  26  and  11  a.m.  on  October  27  at  Buffalo  the 
barometric  effect  increased  very  rapidly  from  +0.01  foot  to  +0.50  foot. 
Between  1  a.m.  and  10  a.m.  on  October  27  the  wind  effect  increased  very 
rapidly  from  —0.07  to  +1.02  feet.  The  two  nearly  simultaneous  impulses, 
one  from  the  barometric  pressures  and  one  from  the  winds,  together  tended  to 
raise  the  water  surface  at  Buffalo  by  1.58  feet.  Actually,  according  to  the 
observed  elevations,  the  water  rose  at  Buffalo  4.65  feet  —  from  elevation 
571.05  at  midnight  at  the  beginning  of  October  27  to  elevation  575.70  at  10 
a.m.  on  October  27  —  at  the  time  when  the  maximum  wind  effect  occurred 
with  a  wind  of  53  miles  per  hour  from  the  southwest. 

The  wind  effect  at  Buffalo  decreased  very  rapidly  from  +1.02  feet  at  10 
a.m.  on  October  27  to  +0.14  foot  at  4  p.m.  on  October  27,  by  which  time 
the  wind  had  died  down  to  27  miles  per  hour  from  the  west.  Note  that  the 
water  surface  (observed  elevations)  fell  at  Buffalo  4.60  feet  —  from  elevation 
575.70  at  10  a.m.  on  October  27  to  elevation  571.10  at  7  p.m.  From  the 
facts  to  which  attention  has  been  called  in  this  paragraph  and  the  preceding 
paragraph  it  is  clear  that  the  great  rise  culminating  in  the  elevation  575.70 
at  10  a.m.  on  October  27  was  largely  an  inertia  effect.  The  barometric 
effect  and  wind  effect  combined,  without  inertia,  would  have  accounted  for 
only  1.58  feet  out  of  the  total  rise  of  4.65  feet. 

That  the  great  rise  was  largely  due  to  inertia  effects  and  was  the  first 
half  wave  of  a  new  very  large  seiche  is  shown  by  the  continuous  graph 
(corrected  elevations).  It  is  such  a  first  half  wave  of  a  new  seiche,  pro- 
duced by  a  new  large  impulse,  that  is  believed  to  produce  the  occasional 
excessively  large  residuals  in  the  daily  corrected  elevations  which  are  caught 
and  rejected  by  the  criterion  for  such  rejections  which  is  set  forth  on  page 
111.  Note  in  table  No.  19,  page  83,  that  the  residual  from  the  five-day 
mean  for  corrected  elevations  for  October  27, 1910,  at  Buffalo  was  —0.39  foot, 


BAROMETRIC  PRESSURES  ON  THE  GREAT  LAKES     123 

and  that  the  corrected  elevation  for  this  day  was  automatically  rejected  by 
the  criterion. 

Compare  the  graphs  for  Cleveland  on  October  27,  1910,  as  shown  on  plate 
16,  with  those  for  Buffalo  on  that  plate.  Note  that  the  barometric  effects 
and  wind  effects  were  much  smaller  at  Cleveland  on  that  day  than  at 
Buffalo.  Note  that  the  minimum  observed  elevation  occurred  at  Cleveland 
at  8  a.m.,  within  2  hours  of  the  maximum  observed  elevation  at  Buffalo,  and 
that  the  maximum  observed  elevation  occurred  at  Cleveland  at  8  p.m., 
within  1  hour  of  being  simultaneous  with  the  minimum  observed  at  Buffalo 
at  7  p.m.  The  rise  of  1.57  feet  at  Cleveland  between  8  a.m.  and  8  p.m. 
corresponded  to  the  fall  of  4.60  feet  at  Buffalo  between  10  a.m.  and  7  p.m. 
This  is  in  accordance  with  the  idea  that  this  was  largely  an  inertia  effect, 
the  early  part  of  a  new  seiche  affecting  the  whole  of  Lake  Erie  and  of  the 
character  indicated  on  page  118.  Note  that  in  such  a  seiche  Cleveland  and 
Buffalo  water  surfaces  should  normally  be  changing  in  opposite  directions, 
and  that  Cleveland  minima  should  be  simultaneous  with  Buffalo  maxima, 
and  vice  versa. 

GENERALIZATIONS. 

In  the  course  of  this  investigation,  during  the  consideration  of  the  many 
details  of  the  evidence  which  it  is  not  feasible  to  set  forth  here,  and  in  the 
efforts  to  develop  an  adequate  theory,  the  writer  has  made  certain  general- 
izations in  regard  to  barometric  effects,  wind  effects,  and  seiches.  These 
generalizations  have  not  been  fully  established  in  some  parts.  Even  in 
such  parts  as  are  fully  established  it  is  not  feasible  to  set  forth  the  con- 
siderations on  which  they  are  based,  except  in  part.  Yet  even  under  these 
circumstances  it  is  desirable  to  give  the  generalizations  as  a  guide  to  others, 
to  be  used  with  such  degree  of  caution  as  may  seem  best  to  them.  The 
generalizations  are  accordingly  given  in  what  follows. 

GENERALIZATIONS  AS  TO  BAROMETRIC  EFFECTS. 

Barometric  effects  upon  the  elevation  of  the  water  surface  at  a  given 
point  are  primarily  proportional  to  the  distance  of  said  point  from  the  center 
of  gravity  of  the  area  of  the  lake  surface.  Note  the  formulae  for  Rv  and  Rn, 
(17),  on  page  16. 

Barometric  effects  at  a  given  point  are  also  dependent  to  a  considerable 
extent  upon  the  shape  of  the  bottom  and  the  configuration  of  the  shore  in 
all  parts  of  the  lake.  This  influence  shows  in  tending  to  make  the  Pw  and 
Pn  proportionality  factors  in  (19),  page  17,  depart  widely  from  unity. 
Note  the  value  of  the  factors  found  in  this  investigation  in  table  No.  16, 
page  70.  It  is  probable  that  these  proportionality  factors  tend  to  be 
much  greater  than  unity,  and  the  barometric  effects  correspondingly  large, 
on  all  lakes  having  long  natural  periods  of  oscillation,  say  more  than  6  hours. 
If  the  natural  period  of  oscillation  is  much  less  than  6  hours,  the  water  will 
tend  to  respond  to  the  rather  slowly  changing  barometric  gradients  in  such 


124  EFFECTS  OF  WINDS  AND  OF 

wise  as  to  maintain  a  moderately  close  approach  to  the  condition  of  equilib- 
rium, in  which  case  the  proportionality  factors  would  tend  to  be  unity. 

The  magnitude  of  the  barometric  effects  on  a  given  lake  is  dependent 
upon  the  position  of  that  lake  in  regard  to  prevailing  storm  tracks.  The 
principal  storm  tracks  for  the  United  States  pass  near  the  Great  Lakes. 
Hence,  there  are  frequent  cases  of  large  and  rapidly  changing  barometric 
gradients  over  these  lakes.  If  lakes  of  the  same  shape  and  size  in  every 
respect  existed  in  some  tropical  region  in  which  fluctuations  in  barometric 
gradients  were  small,  as  a  rule,  the  barometric  effects  would  be  correspond- 
ingly small.  The  barometric  effects  on  Gatun  Lake,  at  the  Panama  Canal, 
are  probably  relatively  small,  for  the  reason  that  the  fluctuations  in  baro- 
metric gradients  are  small. 


GENERALIZATIONS  AS  TO  WIND  EFFECTS. 

The  wind  effects  upon  the  elevation  of  the  water  surface  at  a  given  point 
are  dependent  to  a  small  extent  upon  the  distance  of  that  point  from  the 
nodal  line.  Note  the  function  which  L,  the  distance,  plays  in  formula  (59), 
page  43. 

The  wind  effect  at  a  given  point  is  dependent  largely  upon  the  average 

value  of  the  inverse  cube  of  the  depth,  ~,  between  that  point  and  the  nodal 
line.  Consult  formula  (59) .  This  average  value  is  dependent  mainly  upon 
the  smaller  depths  involved.  Dividing  the  depth  by  10  multiplies  — 

by  1,000.  Hence,  in  making  a  first  estimate  of  the  probable  magnitude  of 
the  wind  effects  at  a  given  point,  attention  should  be  riveted  mainly  upon 
the  shallow  portions  of  the  lakes  which  lie  between  the  point  and  the  nodal 
line. 

The  nodal  line  for  each  direction  of  wind  tends  to  be  located  close  to  the 
shallow  parts  of  a  lake  which  is  unsymmetrical  as  to  depths.  Parts  of  the 
nodal  line  tend  to  cross  the  mouths  of  bays  of  small  depth.  Note  the 
positions  of  the  nodal  lines  on  Lake  Erie,  and  the  several  cases  shown  in 
plates  2,  5,  and  6,  in  which  detached  portions  of  a  nodal  line  cross  bays 
tributary  to  Lake  Michigan-Huron.  There  are  several  other  such  cases 
on  Lake  Michigan-Huron,  which  could  not  be  shown  clearly  on  the  small- 
scale  illustrations  of  this  publication. 

The  wind  effects  tend  to  be  very  large  in  bays  of  small  depth,  both  be- 
cause of  the  small  depths  involved  and  because  of  the  fact  that  cross  return 
currents,  with  a  component  at  right  angles  to  the  wind  direction,  such  as 
are  referred  to  on  page  43,  which  otherwise  would  hold  down  the  wind 
effects,  are  inhibited  by  land  intervening  between  the  bay  and  the  open 
lake.  In  making  a  first  rough  estimate  of  the  wind  effect  at  a  point  on  a 
bay  it  is  important  to  note  carefully  the  extent  to  which  such  cross  currents 
are  inhibited  for  winds  in  each  direction. 


BAROMETRIC   PRESSURES   ON   THE    GREAT  LAKES  125 

The  magnitude  of  the  wind  effects  on  a  given  lake  probably  has  little 
relation  to  the  natural  periods  of  oscillation,  or  seiche  periods,  of  that  lake. 
The  fluctuation  in  wind  impulses  imparted  to  the  lake  by  the  surface  drift 
to  leeward  of  the  water  is  so  rapid  and  so  erratic,  and  the  surface  drift 
normally  remains  nearly  constant  for  so  short  a  period,  except  during  very 
light  winds,  that  the  inertia  effects,  upon  an  average,  probably  tend  about  as 
frequently  to  act  counter  to  the  wind  impulse  as  to  act  in  cooperation  with 
it.  The  one  exception  to  the  foregoing  generalization  is  that  the  first  half 
wave  of  a  new  extremely  large  seiche  tends  frequently  to  be  such  as  to 
correspond  to  an  exaggerated  wind  effect. 

GENERALIZATIONS  AS  TO  SEICHES. 

The  initial  impulse  or  impulses  which  start  a  seiche  in  the  Great  Lakes 
are  probably  much  more  frequently  due  to  the  wind  than  to  the  barometric 
gradients.  The  wind  impulses  are  in  general  much  more  sudden  than  the 
barometric  impulses.  They  are  certainly  much  more  frequent.  The 
barometric  impulses  are  far  from  negligible,  however,  on  the  Great  Lakes. 
The  impulse  starting  a  seiche  is  clearly  traceable  to  the  barometric  gra- 
dients in  some  cases.  The  barometric  gradients  are  probably  especially 
effective  in  starting  the  Strait  of  Mackinac  seiches  referred  to  on  page  119. 

The  development  or  non-development  of  seiches  on  a  given  lake  is  largely 
dependent  upon  the  depth  of  the  lake,  upon  the  uniformity  of  depth,  and 
upon  the  configuration  of  the  shore.  The  greater  the  depth  of  the  lake 
the  smaller  will  be  the  role  played  by  friction  and  the  smaller  the  true 
damping  of  the  seiche.  The  more  uniform  the  depth  over  the  greater  part 
of  the  lake  the  more  will  the  lake  tend  to  act  as  a  single  large  seiche  area  and 
the  larger  and  more  persistent  will  the  seiches  tend  to  be.  Similarly,  the 
more  regular  the  shore  as  seen  in  horizontal  projection  the  more  will  the  lake 
tend  to  act  primarily  as  one  large  seiche  area,  with  correspondingly  large 
and  persistent  seiches. 

Normally,  a  lake  surface  has  several  natural  periods  of  oscillation  or 
seiche  periods.  Each  seiche  period  and  each  method  of  oscillation  as  a 
seiche  (lengthwise  or  crosswise)  pertains  to  what  may  be  called  a  seiche 
area.  Each  seiche  area  is  limited  either  (a)  by  portions  of  the  shore  of  the 
lake  or  (6)  by  a  belt  in  which  there  is  a  steep  slope  in  the  bottom  of  the  lake 
separating  areas  of  decidedly  different  depths.  At  boundaries  of  a  seiche 
area  of  class  (a)  there  is  no  transmission  of  the  oscillation  beyond  the 
boundary.  There  is  simply  absorption  and  reflection.  At  boundaries  of 
a  seiche  area  of  class  (6)  there  is  much  more  or  less  effective  reflection,  de- 
pending upon  the  steepness  of  the  slope  of  the  bottom  and  the  amount  of 
the  total  change  of  depth,  as  well  as  upon  the  shape  of  such  boundary  as 
seen  in  horizontal  projection.  To  the  extent  that  such  reflection  takes 
place,  the  belt  acts  as  a  boundary.  There  is  also  at  boundaries  of  class  (6), 
in  addition  to  absorption  of  energy  due  to  local  movements,  a  decided 
tendency  for  a  part  of  the  wave  to  be  transmitted  across  the  boundary  into 


126  EFFECTS   OF   WINDS   AND    OF 

the  next  seiche  area,  where  it  becomes  a  forced  oscillation  for  the  time  being 
and  tends  to  set  up  a  new  seiche  of  the  period  natural  to  that  area  or  to 
modify  such  a  seiche  which  may  already  be  there. 

In  each  seiche  area  the  seiche  period  is  dependent  upon  the  depth  of  the 
water  and  the  principal  dimensions  of  the  area. 

The  apparent  damping  of  seiches  is  dependent  upon  two  things:  (1)  the 
true  damping,  due  to  friction,  and  (2)  the  transmission  from  the  seiche  area 
in  question  into  other  adjacent  seiche  areas. 

The  true  damping,  due  to  friction,  is  of  course  dependent  primarily  upon 
the  depth  of  the  water.  In  deep  water  the  friction  is  smaller  in  proportion 
to  the  total  energy  involved  in  the  seiche,  and  hence  the  damping  is  small. 

The  apparent  damping  by  transmission  from  the  seiche  area  under  con- 
sideration into  adjacent  seiche  areas  tends,  of  course,  to  be  greater  (1)  if 
the  seiche  area  in  question  is  bounded  largely  by  other  seiche  areas  rather 
than  by  shores,  (2)  if  the  adjacent  seiche,  areas  are  large  in  comparison  with 
the  seiche  area  under  consideration,  and  (3)  if  the  seiches  in  progress  in  the 
adjacent  areas  are  small  relatively  to  those  in  the  area  under  consideration. 
If  an  adjacent  seiche  area  has  in  it  at  a  given  time  a  large  seiche,  there  is  a 
tendency  for  the  net  transmission  of  energy  across  the  boundary  to  be  such 
as  to  increase  the  seiche  in  the  area  under  consideration.  In  general,  there 
will  be  a  transmission  in  progress  in  both  directions  across  the  boundary,  one 
tending  to  dissipate  the  seiche  in  the  area  under  consideration  and  the  other 
to  build  it  up  at  the  expense  of  the  adjacent  seiche.  In  general,  if  an  un- 
usually large  seiche  is  started  in  one  seiche  area  of  several  in  a  lake,  the  ap- 
parent damping  in  that  area  will  be  very  rapid  at  first,  until  enough  energy 
has  been  transmitted  across  to  adjacent  seiche  areas  to  build  sufficiently 
large  seiches  there  to  make  the  net  exchange  of  energy  across  the  seiche  area 
boundaries  nearly  zero.  Thereafter  the  apparent  damping  will  be  mainly 
true  damping,  due  to  friction,  and  therefore  relatively  slow. 

From  the  foregoing  it  is  clear  that  if  a  lake  has  a  compact  area  with 
regular  shores,  has  nearly  uniform  depths,  and  has  these  matters  so  related 
that  the  whole  lake  is  composed  of  only  one  or  a  few  seiche  areas,  large  and 
persistent  seiches  will  occur  in  that  lake.  On  the  other  hand,  if  a  lake  has 
a  very  irregular  shore  and  a  straggling  area,  if  its  depth  varies  greatly  in 
different  parts,  and  if  these  matters  are  so  related  that  the  whole  lake  is 
broken  up  into  many  seiche  areas  none  of  which  predominates  in  size  over  the 
others,  the  seiches  in  that  lake  will  in  general  be  small,  and  each  new  un- 
usually large  seiche  will  be  rather  promptly  reduced  to  moderate  size  by 
transfer  of  energy  across  seiche-area  boundaries. 

Consider  the  contrast  between  Lake  Erie  and  Lake  Michigan-Huron 
in  the  characteristics  indicated  in  the  preceding  paragraph.  Use  plates  2, 
5,  and  6  in  visualizing  the  contrast. 

As  a  whole,  Lake  Erie  is  comparatively  compact,  with  regular  shores. 
Over  more  than  one-half  of  its  area  the  bottom  is  almost  as  flat  as  a  floor, 
with  depths  varying  only  from  11  to  14  fathoms  for  120  miles  along  its  axis. 


BAROMETRIC  PRESSURES  ON  THE  GREAT  LAKES     127 

It  is  broken  up  into  only  a  few  seiche  areas.  There  are  probably  only  three 
important  seiche  areas  in  Lake  Erie:  (1)  the  area  of  depth  greater  than  20 
fathoms  in  the  eastern  part  of  the  lake;  (2)  the  continuous  area  of  more 
than  10  fathoms  depth  extending  from  a  point  near  the  eastern  end  of  the 
lake  (marked  by  three  stars  on  plate  2)  nearly  to  Sandusky  and  Point  Pelee, 
near  the  western  end  of  the  lake;  and  (3)  the  part  of  the  western  end  of  the 
lake  which  is  largely  cut  off  from  the  main  lake  by  Point  Pelee  and  a  chain 
of  islands  including  Pelee  Island  and  Kelley's  Island.  The  first-mentioned 
seiche  area  is  included  within  the  second-mentioned. 

In  contrast,  consider  Lake  Michigan-Huron.  It  has  a  straggling  area 
bounded  by  extremely  irregular  shores  interrupted  once  by  the  whole  of  the 
lower  peninsula  of  Michigan  and  again  by  the  peninsula  and  islands  which 
in  part  cut  off  Georgian  Bay  and  North  Channel  from  the  main  portion  of 
the  lake.  Lake  Michigan-Huron  is  probably  broken  up  into  at  least  8 
seiche  areas,  which  are  all  of  primary  importance.  The  writer  estimates 
these  areas,  by  examination  of  the  Lake  Survey  charts,  to  be  as  enumerated 
below: 

(1)  That  part  of  Lake  Michigan  south  of  latitude  44°,  in  which  the  depth  is  more  than  50 
fathoms  and  less  than  90.     Note  that  the  regularity  of  this  area  is  broken  in  its  middle 
portion  by  a  small  oval,  within  which  the  depth  is  more  than  90  fathoms,  and  another  oval 
near  it,  within  which  the  depth  is  less  than  50  fathoms. 

(2)  That  northern  part  of  Lake  Michigan  extending  from  a  point  north  of  latitude  44° 
nearly  as  far  north  as  the  main  entrance  to  Green  Bay,  in  which  the  depth  is  more  than 
90  fathoms. 

(3)  Green  Bay. 

(4)  That  northeastern  part  of  Lake  Huron  which  is  more  than  50  fathoms  deep,  ex- 
tending in  a  long  area  of  gradually  increasing  width  from  latitude  46°  and  longitude  84° 
southeastward  to  latitude  44i°  and  longitude  82°. 

(5)  North  Channel,  the  long  narrow  area  at  the  extreme  north  end  of  Lake  Huron, 
separated  from  the  main  portion  of  the  lake  by  Manitoulin  Island. 

(6)  Georgian  Bay. 

(7)  The  unnamed  bay  which  is  a  southern  extension  of  Lake  Huron  and  which  terminates 
at  Port  Huron  and  the  entrance  of  the  St.  Claire  River. 

(8)  SaginawBay. 

So  far  as  there  is  a  7-hour  seiche  through  the  Strait  of  Mackinac,  as  dis- 
cussed on  pages  119-120,  the  whole  of  Lake  Michigan-Huron  is  acting  as  a 
single  seiche  area. 

In  accordance  with  the  contrast  between  Lake  Erie  and  Lake  Michigan- 
Huron,  to  which  attention  has  just  been  called,  the  observations  indicate 
that  the  seiches  on  Lake  Erie  are  much  larger  in  range,  as  a  rule,  than  those 
on  Lake  Michigan-Huron ;  that  the  Lake  Erie  seiches  persist  with  a  relatively 
large  range  for  several  days  at  a  time;  and  that  on  Lake  Michigan-Huron, 
though  very  large  seiches  sometimes  occur,  such  seiches  are  apparently 
damped  down  to  a  small  range,  as  a  rule,  after  only  a  few  oscillations. 
The  prevailing  condition  at  each  gage  on  Lake  Michigan-Huron  seems  to  be 
that  small  seiches  are  in  progress,  seldom  disappearing  completely  and 
seldom  large  and  regular  enough  to  make  it  easy  to  detect  the  seiche  periods. 


128  EFFECTS  OF  WINDS  AND  OF 

This  last  statement  is  cautiously  made,  because  the  limits  of  this  investi- 
gation have  made  it  impossible  to  study  the  seiches  in  Lake  Michigan-Huron 
with  the  same  care  that  was  given  to  Lake  Erie  seiches. 

GENERALIZATION  AS  TO  PREVAILING  CONDITIONS  ON  THE 
GREAT  LAKES. 

Apparently,  if  one  is  to  appreciate  fully  the  meaning  of  the  continuous 
record  from  a  gage  recording  elevations  of  the  water  surface  at  any  place 
in  the  Great  Lakes,  he  must  have  the  general  conception  stated  in  the 
following  long  paragraph,  and  must  appreciate  that  the  gage  is  in  one  of  the 
several  seiche  areas  of  that  lake. 

At  the  beginning  of  any  hour  on  any  one  of  the  Great  Lakes  there  are  one 
or  more  seiches  in  progress  in  each  of  the  several  seiche  areas  of  that  lake, 
and  all  oscillations  (seiches)  are  taking  place  about  a  certain  equilibrium 
surface  which  is  not  horizontal  but  which  has  a  certain  position  and  slopes 
fixed  by  the  winds  and  barometric  gradients  operating  on  the  lake  at  that 
instant.  During  that  hour  changes  occur  in  the  direction  and  velocity  of 
the  winds  and  in  the  barometric  gradients.  These  changes  produce  cor- 
responding changes  in  the  position  and  slopes  of  the  equilibrium  surface 
about  which  all  oscillations  due  to  inertia  (seiches)  take  place.  The  changes 
in  the  equilibrium  surface  produce  new  forced  oscillations  in  every  seiche 
area.  The  forced  oscillations  develop  new  seiches  with  the  natural  period 
of  each  seiche  area,  which  new  seiches,  being  superposed  on  the  seiches 
already  there,  modify  them,  sometimes  mainly  in  phase,  sometimes  to  par- 
tially stop  them,  but  as  a  rule  to  increase  their  range.  From  each  seiche 
area  forced  oscillations  are  sent  continuously  into  adjacent  seiche  areas, 
tending  to  develop  in  these  areas  new  seiches,  which  are  superposed  on  the 
old,  as  indicated  in  the  preceding  sentence.  In  general,  whenever  a  change 
occurs  in  the  equilibrium  surface  —  when  there  is  a  new  set  of  wind  and 
barometric  impulses  over  a  lake  —  there  will  be  a  considerably  larger  seiche 
produced  in  some  one  of  the  various  seiche  areas  than  in  the  others.  This 
may  come  about  because  (a)  the  new  impulse  happens  to  be  timed  to  cor- 
respond in  the  period  of  its  change  to  the  natural  period  of  that  seiche  area, 
or  (6)  the  new  impulse  may  happen  to  reinforce  a  particular  wave  of  a  seiche 
in  progress  in  that  area  and  so  to  decidedly  increase  its  range,  or  (c)  the 
forced  oscillations  sent  in  from  the  adjacent  areas  may  happen  to  coincide 
with  each  other  and  with  a  seiche  in  progress  and  so  to  decidedly  increase 
its  range.  The  exchange  of  energy  between  different  seiche  areas  will  tend 
to  produce  a  very  much  greater  apparent  damping  of  this  particular  large 
seiche  than  of  the  other  seiches  in  progress  and  tend  to  bring  about  a  nor- 
mal distribution  of  seiche  range  to  the  different  areas.  As  time  progresses 
there  will  be  continual  modification  of  seiches  by  new  wind  and  baromet- 
ric impulses,  usually  tending  toward  an  increase  in  range  of  the  seiches. 
The  true  damping  by  friction,  on  the  other  hand,  tends  continually  to 
reduce  the  range  of  the  seiches.  The  net  effect  of  these  two  counter  in- 


BAROMETRIC  PRESSURES  ON  THE  GREAT  LAKES     129 

fluences  on  seiche  range  is  to  cause  the  seiche  range  in  general  to  increase 
during  periods  of  increasing  intensity  of  wind  and  barometric  impulses,  to 
decrease  rather  rapidly  during  periods  of  decreasing  intensity  of  wind  and 
barometric  impulses,  and  still  to  decrease  but  slowly  during  periods  when 
such  intensity  is  approximately  constant.  One  must  see  all  these  things  to 
appreciate  fully  the  meaning  of  a  gage  record,  in  addition  to  seeing  in  it  the 
fluctuations  of  elevation  of  the  mean  lake  surface  due  to  inflow  from  the 
lake  above,  outflow  to  the  lake  below,  run-off  from  the  surrounding  land, 
rainfall  on  the  lake  surface,  and  evaporation  from  the  lake  surface. 

POSSIBLE  APPLICATIONS  OF  RESULTS  OF  THIS 
INVESTIGATION. 

The  outcome  of  this  investigation  is  a  more  accurate  knowledge  than  was 
heretofore  available  of  the  wind  effects  and  the  barometric  effects  upon  the 
elevation  of  the  water  surface  at  any  given  point  on  the  Great  Lakes  or  on 
any  free  water  surface,  and  some  increase  in  the  available  knowledge  of 
seiches  and  of  their  relation  to  accurate  determinations  of  the  mean  elevation 
of  the  water  surface  from  a  given  gage  record.  Consider,  now,  very  briefly, 
some  of  the  possible  applications  of  the  results  of  this  investigation. 

APPLICATION  TO  A  STUDY  OF  LAWS  OF  EVAPORATION. 

As  stated  on  pages  1-4,  the  investigation  here  reported  upon  is  a  part  of  a 
larger  investigation  having  for  its  purpose  determining  the  laws  of  evapo- 
ration from  large  water  surfaces,  such  as  the  surfaces  of  lakes  and  rivers.  To 
attain  this  end,  it  is  proposed  to  consider  each  of  the  Great  Lakes  in  turn  as 
an  evaporation  pan  and  to  evaluate  from  day  to  day  (1)  the  change  of  con- 
tent, (2)  the  income,  and  (3)  the  outgo,  including  evaporation.  To  ac- 
complish the  purpose  it  is  necessary  to  evaluate  the  change  of  content 
from  day  to  day  with  great  accuracy.  That  change  of  content  is  measured 
by  the  change  in  elevation  from  day  to  day  of  the  mean  surface,  of  the  whole 
lake.  Each  recording  gage  measures  the  change  in  elevation  of  the  surface 
of  the  lake  at  the  point  at  which  the  gage  is  located.  Formerly  the  only 
feasible  way  to  fix  the  elevation  of  the  mean  surface  of  the  lake  was  to  take 
it  as  equal  to  the  mean  of  the  elevations  at  the  two  or  three  points  on  the 
lake  at  which  first-class  gages  were  operating.  Now,  as  an  outcome  of  this 
investigation,  the  elevation  of  the  mean  surface  of  the  whole  lake  may  be 
determined  on  any  day  by  applying  the  known  corrections  for  wind  effects 
and  barometric  effects  at  any  gage  to  the  observed  elevation  for  that  gage. 
The  proper  weighted  mean  for  the  several  gages  may  be  taken  and  the  few 
abnormal  values  may  be  detected  and  rejected  by  a  definite  criterion.  In 
the  place,  then,  of  the  former  values  of  mean  elevation  of  the  whole  lake 
surface,  of  a  certain  degree  of  accuracy,  one  has  now  available  values  of  a 
much  higher  degree  of  accuracy.  The  increase  in  accuracy,  which  operates 
to  increase  the  possible  accuracy  of  the  evaporation  investigation,  may  be 
seen  (a)  by  comparison  of  the  two  sets  of  residuals  in  tables  Nos.  19  to  25, 


130  EFFECTS   OF   WINDS   AND    OF 

(6)  in  the  probable  errors  shown  in  table  No.  28,  page  110,  and  (c)  in  the 
discussion  of  these  probable  errors  on  pages  111-113.  It  appears  that  from 
observations  at  Mackinaw  alone  the  mean  elevation  of  the  whole  of  Lake 
Michigan-Huron  on  any  day  may  possibly  be  determined  with  a  probable 
error  less  than  ±0.010  foot.  From  the  three  stations  Milwaukee,  Harbor 
Beach,  and  Mackinaw  together,  this  probable  error  may  be  reduced  still 
more.  It  appears  that  the  change  in  elevation  of  the  mean  surface  of  the 
whole  of  Lake  Michigan-Huron  in  one  day  may  possibly  be  determined 
with  a  probable  error  less  than  ±0.007  foot  —  an  accuracy  hitherto  un- 
attainable. 

The  evaporation  from  Lake  Michigan-Huron  in  a  single  day,  measured  in 
depth  of  water  taken  off  the  whole  surface  of  the  lake,  probably  varies 
between  limits  which  are  approximately  0.000  foot  and  0.021  foot.  The  new 
accuracy  now  attainable  in  determining  the  fluctuation  in  the  elevation  of 
the  mean  surface  of  the  lake  from  day  to  day  evidently  makes  the  proposed 
method  of  measuring  the  small  variations  in  evaporation  much  more  power- 
ful than  it  otherwise  would  be.  Heretofore  the  method  seemed  to  be  barely 
within  the  range  of  possibility.  Now  it  seems  to  be  certain  that  the  method 
will  succeed.  It  certainly  will  succeed  so  far  as  success  hinges  on  the 
evaluation  from  day  to  day  of  the  total  content  of  the  lake. 

APPLICATION  TO  REGULATION  OF  THE  GREAT  LAKES. 

The  vision  is  gradually  taking  form  that  the  elevation  of  the  water  surface 
of  each  of  the  Great  Lakes  must  be  regulated  by  movable  dams,  or  the 
equivalent,  for  the  benefit  of  navigation,  for  the  benefit  to  be  derived  in 
connection  with  the  development  of  power  in  hydro-electric  stations  on  the 
connecting  streams,  and  to  provide  for  the  increased  use  of  the  water  from 
Lake  Michigan-Huron  for  sanitary  purposes  at  Chicago.  It  is  now  clear 
that  the  return  to  the  people  of  the  United  States  from  such  regulation 
would  be  many  times  its  cost. 

When  that  regulation  becomes  a  reality,  as  it  certainly  will  in  due  time, 
it  will  then  become  important  to  detect  each  fluctuation  in  elevation  of  each 
lake  surface  as  soon  as  possible  after  it  occurs  and  as  accurately  as  possible. 
Each  fluctuation  must  be  detected  soon  after  it  occurs  in  order  that  the 
desirable  change  in  regulation  may  be  made.  It  must  be  detected  accurately, 
because  the  total  range  within  which  the  regulation  must  operate  is  small, 
and  the  means  of  regulation,  the  movable  dams  or  the  equivalent,  will 
produce  but  very  slow  alterations  in  lake  elevations.  The  quantities  to  be 
dealt  with,  for  the  desired  valuable  regulation,  are  hundredths  of  feet  of  ele- 
vation rather  than  tenths.  The  promptness  and  accuracy  with  which  all 
fluctuations  may  be  recognized  is  greatly  increased  by  the  outcome  of  this 
investigation.  This  is  indicated  in  part  by  the  comments  on  accuracy  made 
in  connection  with  the  proposed  application  to  the  study  of  evaporation. 
It  may  be  emphasized  from  another  point  of  view  by  noting  that  there  are 


BAEOMETRIC   PRESSURES   ON   THE    GREAT   LAKES  131 

frequent  periods  of  10  days  or  more  on  which  the  observed  elevations  at  a 
gage  are  continuously  too  high  or  continuously  too  low  to  represent  the 
elevation  of  the  mean  lake  surface.  As  extreme  cases,  note  the  facts  for 
the  Cleveland  gage  in  August  1910  and  for  the  Milwaukee  gage  in  July 
1911. 

At  Cleveland  (see  table  No.  20,  page  85,  table  No.  26,  page  108,  and 
plates  8  and  9),  in  August  1910  the  correction  for  barometric  effect  was 
+  on  25  days  out  of  31.  The  maximum  correction  for  barometric  effect 
was  +0.24  foot  on  August  25.  On  the  5  days  August  21-25  the  average 
correction  was  +0.18  foot.  For  the  whole  month  of  August  the  corrected 
elevation  was  0.06  foot  greater  than  the  observed. 

At  Milwaukee  (see  table  No.  21,  page  88,  table  No.  27,  page  109,  and 
plates  11  and  12),  in  July  1911  the  correction  for  barometric  effect  was  + 
on  28  days  out  of  31.  The  maximum  correction  for  barometric  effect  was 
+0.30  foot  on  July  25.  On  the  7  days  July  20-26  the  average  correction 
was  +0.15  foot.  For  the  whole  month  of  July  1911  the  corrected  elevation 
was  0.09  foot  greater  than  the  observed. 

The  results  of  this  investigation,  in  the  form  of  corrections  for  wind 
effects  and  barometric  effects,  and  detection  of  abnormal  values  by  certain 
criteria,  should  be  applied  to  regulation  of  the  Great  Lakes,  (a)  directly,  to 
furnish  a  more  accurate  knowledge  than  would  otherwise  be  possible,  day  by 
day,  of  the  actual  mean  elevation  of  the  whole  surface  of  each  lake,  and  there- 
fore of  the  total  water  content  of  the  lake,  and,  (6)  indirectly,  by  enabling 
the  laws  of  evaporation  and  of  run-off  into  the  lake  to  be  determined  and 
understood  in  such  wise  that  a  forecast  of  the  total  yield  of  water  to  each 
lake  for  weeks  and  possibly  months  in  advance  may  be  made  with  much 
greater  accuracy  than  would  otherwise  be  possible. 


APPLICATION  TO  DETERMINATION  OF  MEAN  SEA-LEVEL  AND  TO 
PRECISE  LEVELING. 

Elevations  determined  by  precise  leveling  are  referred  to  mean  sea-level 
by  means  of  observations  taken  at  tide  gages.  The  mean  sea-level  as  fixed 
by  the  observations  at  a  given  gage  is  in  error  by  an  amount  dependent  on 
the  configuration  of  the  shores  and  the  bottom  in  the  surrounding  region 
and  upon  the  prevailing  winds.  The  method  and  constants  now  available 
will  enable  one  to  compute  the  necessary  correction  to  be  applied  to  the 
observed  mean  sea-level  to  eliminate  the  wind  effect  and  so  to  obtain  the 
true  mean  sea-level.  Such  a  necessary  correction  may  be  small  at  certain 
gages.  It  is  important  to  prove  it  to  be  small  in  such  cases.  The 
corrections  are  probably  large  enough  at  some  gages  to  predominate  over 
the  accumulated  errors  in  the  precise  leveling  for  hundreds  of  miles  from 
the  gages. 

The  corrections  for  barometric  effect  should  be  applied  similarly  in  con- 
nection with  the  determination  of  mean  sea-level. 


132  EFFECTS  OF  WINDS  AND  OF 

A  concrete  idea  of  the  possible  errors  inherent  in  the  present  determina- 
tions of  mean  sea-level  and  their  relation  to  the  errors  of  precise  leveling 
may  be  secured  as  follows:  Note  that  at  the  bottom  of  tables  Nos.  26  and 
27  the  mean  elevation  as  observed  for  a  whole  season  agrees  within  0.03  foot 
with  the  corrected  elevation  for  the  season  in  every  case  except  at  Buffalo. 
At  Buffalo,  however,  the  mean  of  the  observed  elevations  for  the  three 
months  of  1909  differs  by  0.11  foot  from  the  mean  of  the  corrected  elevations, 
and  for  the  5  months  of  1910  the  difference  is  0.10  foot.  In  other  words, 
the  omission  of  the  wind  and  barometric  corrections  at  Buffalo  leaves  an 
avoidable  error  of  0.10  foot  in  the  derived  elevation  of  the  water  surface 
from  the  5  months  of  observation. 

The  probable  error  of  a  difference  of  elevation  of  two  points,  determined 
by  the  precise  leveling  of  the  Coast  and  Geodetic  Survey,  is  about  ±0.7 
millimeter  into  the  square  root  of  the  distance  between  the  two  points  in 
kilometers  measured  along  the  line  of  leveling.  In  other  words,  an  error  of 
0.03  foot  =9.1  millimeters  has  an  even  chance  of  occurring  in  169  kilometers 
=  105  miles  of  leveling  from  a  tide  gage.  The  error  of  0.10  foot  referred  to 
above  in  connection  with  the  5  months  observation  at  Buffalo  has  an  even 
chance  of  occurring  in  about  1,800  kilometers  =  1,100  miles  of  precise  level- 
ing. This  statement  neglects  the  possible  small  systematic  errors  in  the 
leveling  and  also  the  strengthening  of  the  level  line  by  its  connection  with  a 
net  of  level  lines. 

Judging  from  a  study  of  the  charts  and  his  present  knowledge  of  the  laws 
controlling  wind  effects,  the  writer  estimates  that  the  wind  effects  are  as 
persistently  large  and  of  one  sign  at  Cedar  Keys,  Florida,  as  at  Buffalo. 
One  of  the  gages  at  which  mean  sea-level  was  determined  for  use  in  con- 
nection with  precise  leveling  was  at  Cedar  Keys.  Similarly,  the  writer 
estimates  that  at  Sandy  Hook,  New  Jersey,  near  New  York  City,  the  chance 
of  error  due  to  the  omission  of  corrections  for  wind  effects  and  barometric 
effects  is  not  much  less  than  that  at  Buffalo  and  is  certainly  greater  than  that 
at  the  other  four  stations  of  tables  Nos.  26  and  27.  Hence,  even  though 
mean  sea-level  is  determined  from  an  average  of  several  years  of  observa- 
tion at  Sandy  Hook,  it  is  possibly  in  error  by  an  amount  exceeding  the  error 
accumulated  in  the  precise  leveling  over  hundreds  of  miles.  It  should  not 
be  overlooked  in  this  connection  that  the  prevailing  winds  and  the  prevail- 
ing barometric  gradients  tend  to  be  seasonal,  to  be  repeated  each  year,  and 
that  therefore  the  taking  of  a  mean  for  several  years  is  of  only  moderate 
effectiveness  in  reducing  the  error  in  the  mean.  The  monthly  values  of  mean 
sea-level  at  various  tide  gages  support  the  statement  by  showing  a  seasonal 
variation,  as  a  rule,  and  thereby  incidentally  indicating  that  the  wind 
effects  and  barometric  effects  are  certainly  decidedly  appreciable  in  the 
monthly  means. 


BAROMETRIC   PRESSURES   ON  THE   GREAT   LAKES  133 

APPLICATION  TO  DETERMINATION  OF  TILTING  OF  THE  GREAT 
LAKES  REGION. 

From  the  evidence  derived  from  gages  operated  over  long  periods  or 
during  widely  separated  years,  at  various  points  of  Lake  Michigan-Huron, 
the  eminent  geologist,  G.  K.  Gilbert,  determined  that  the  whole  region 
covered  by  this  lake  is  slowly  tilting  to  the  southwestward,  and  secured  a 
determination  of  the  rate  of  tilting.  This  determination  would  obviously 
be  strengthened  if  the  corrections  for  wind  effects  and  for  barometric  effects 
at  the  gages  were  applied  by  using  the  methods  and  constants  made  available 
by  this  publication.  This  investigation  by  Gilbert  was  published  as  a  part 
of  the  Annual  Report  of  the  U.  S.  Geological  Survey  for  1896-97,  part  II, 
pages  595-647,  under  the  title  Recent  Earth  Movement  in  the  Great  Lakes 
Region. 

The  rate  of  tilting  as  derived  was  .0042  foot  per  mile  per  century  —  an 
exceedingly  small  rate  of  change.  The  conclusion  was  derived  from  apparent 
changes  of  relative  elevation  of  the  water  surface  as  measured  at  different 
gages  on  Lakes  Michigan-Huron,  Erie,  and  Ontario  in  different  years. 
The  amounts  of  change  involved  are  of  the  order  of  0.1  to  0.2  foot  in 
a  period  of  20  to  40  years.  Evidently,  when  such  small  changes  are  in 
question  there  is  more  chance  of  securing  the  necessary  accuracy  if  cor- 
rections as  large  as  those  shown  in  tables  Nos.  19  to  23,  pages  80-96  of 
this  publication,  for  barometric  effects  and  wind  effects,  are  taken  into 
account.  Gilbert  limited  his  deductions  largely  to  days  on  which  there  was 
little  wind.  But  an  inspection  of  the  computations  made  in  connection 
with  the  present  investigation  shows  that  the  barometric  corrections  are  by 
no  means  negligible  on  the  Great  Lakes  on  days  of  little  wind.  Gilbert 
himself  saw  the  desirability  of  such  corrections,  as  shown  by  various  state- 
ments in  his  paper. 

The  deductions  of  Gilbert  are  probably  correct  in  the  main.  But  a  new 
investigation  based  on  observed  elevations  of  water  surface  corrected  for 
wind  effects  and  barometric  effects  would  have  greater  accuracy  and  is 
desirable.  Such  a  new  investigation  might  be  in  part  a  recomputation 
from  the  data  which  were  used  by  Gilbert.  Much  new  data  of  a  high  degree 
of  accuracy,  in  the  form  of  observations  made  in  the  24  years  since  Gilbert's 
paper  was  written,  are  also  available. 


PLATE  1 


CONTOURS 

ON 
WATER  SURFACE 


JI3O 
NODAL  LINE 


.0339 


.0452 


•0000   .0113    .O226 
Isobars  and  contours  on  water  surface  of  Lake  Erie,  August  5,  1910. 


PLATE  2 


LAKE  ERIE 


N.L.     Nodal  lines_ 
r^~^-\  Depths  in  fathoms 
Seiche  limrts 


88*'          87°  86°       '   85°          84°          «3°  8a°          81°  8O°  79°.          78° 


-f-    Barometric  points 

•Jfc    Center  of  gravity  of  Lakes  Michigan-Huron 


Nodal  lines  and  depths  in  Lake  Erie;  barometric  points  around  the  Great  Lakes. 


PLATE  3 
PRECEDING  DAY  CURRENT  DAY 


8  1 

>M.R 

2               i 
M.          A 

3 
M.    h 

£ 
J.             R 

}     1 

M.  R 

2               £ 
M.         A 

3 

M.  r 

€ 
4.             R 

3            1 

M.R 

2               £ 
M.          AJ 

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w 

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X 

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x 

t 

wo    w 

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x 

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x 

x 

t 
1  w 

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x 

T 

W3      w 

B0    NO  LAG 


B,     NO  LAG 


Bo   NO  LAG 


B3    NO  LAG 


B0  4-h.LAG 


B,    4h.  LAG 


4h.LAG 


B3   4h.  LAG 


Relation  of  barometric  change  to  rise  of  water  surface.    See  text  under  subheading, 
"  Form  of  Observation  Equations  for  Barometric  Effects." 


PLATE  4 


FlG.3  Bottom^      FlG.4    FlG.5 

Wind  effects,  disturbed  water  surface,  and  currents. 


PLATE  5 


N.L. 


rfsSL 

T        \, 

\MACKIJW 

— 3r 


N.L. 


MILWAUKEE 


Nodal  lines 
Depths  in  fathoms. 


N.L 


Nodal  lines  and  depths  in  Lake  Michigan. 


PLATE  6 


Nodal  lines 


Depths  in  fathoms 


Nodal  lines  and  depths  in  Lake  Huron. 


PLATE? 


572.20 


.60 


572.50 


JUNE  1,1910      JUN 


JUN 


JUN 


I  16 


JUN 


E  2123 


JUNE  23         JUN 


E  28 


JUL 


JUL 


JUL* 


13  15 


572.20 


\ 


\    .50 


V 


572.20 


•— Rainfall  +  inflow  —  outflow 

--Observed      Corrected  for  barometric 

and  wind  effects       •••Record  missing 


Elevations  of  water  surface,  Lake  Erie,  June  1-July  15,  1910. 


PLATE  8 


.50 


572;20 


K 


4 

.40 
.30 


/ 


V.      / 


\BUFEALO 


\/ 


\      XX 

--V-/-0- 


/ 


JO 


\ 


572.20 


JULY  15, 1910     JUl 


20 


JUL 


25  JUL 


30 


AUG 


4      6 


5    II 


AUG 


26  28 


72.20 


/ 


Observed 

Corrected  for  barometric 
and  wind  effects 

Rainfall  4-  inflow  —  outflow 
Record  missing 


.10 


x       / 


.30 


572.20 


Elevations  of  water  surface,  Lake  Erie,  July  15-August  28,  1910. 


PLATE  9 


• -Observed Corrected  for  barometric  and  wind  effects 

Rainfall  +  inflow -outflow    Record  missing 

Elevations  of  water  surface,  Lake  Erie,  August  28-October  11,  1910. 


PLATE  10 
CT.1  1,1910          OCT.  16                   OCT.ZI                    OCT.  26                   OCT. 

/'•• 

^ 

572.00 

//'•••'•• 

/>•<.../ 

r^^^ 

j^7\.9Q 

f        -:.^-z=. 

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X/^Sa 

CLEVELAND 

\  /  \/ 

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\!      - 

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Observed 

f, 

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i 

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-outflow 

1  1 

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Elevations  of  water  surface,  Lake  Erie,  October  11-31,  1910. 


PLATE  11 


.10 


HARBOR  BZ'ACH 


579.90 


MILWAUKEE 


579.90 


JUNEI,I9II     JUNES 


JUN 


Ell 


JUN 


JUN 


JUNE  23   •    JUNI 


28 


JUL 


JUU 


JUL 


13    15 


.80 


580.10 


\  X 


579.99. 


MLLWAUKE. 


3.00 


.Corrected  for  barometric  and  wind  effects Observed 

-Rainfall  +  inflow  -outflow  Record  missing 


Elevations  of  water  surface,  Lake  Michigan-Huron,  June  1-July  15,  1911. 


PLATE  12 


.10 


HARBOR  JSEACIf    I 


580.00 


\    / 


MILWAUKEE 


.70 


579.90 


80 


LAKE  MICL  1GAN-HURQN  MEAFT 


JULY  15,1911       JUIY  20 


25 


JUi: 


30 


AUG 


4       6 


AUG    6 


AU 


3  II 


AUC 


16 


AUG 


21 


AUG 


26    28 


HARBOR  &  EACH 


579.90 


579.90 


".BO 


LAK  ?M2CHIGA7r-i  lURCBST. 


Observed 

Corrected  for  barometric  and  wind  effects 

•  Rainfall  +  inflow   —  outflow 

Elevations  of  water  surface,  Lake  Michigan-Huron,  July  15- August  28,  1911. 


MACKINAW 


PLATE  13 


579.70 


.50 


579. 70^ 


2JARB  OR  &EACJ1 


580.00 


MILWAUKEE 


579.90 


579.70 


AUG.  28        SEPT.  2 


SEPT. 


7      19 


SEPT  19         SEP 


24 


SEP 


29 


MACKINAW 


579.70 


.60 


579.60 


Elevations  of  water  surface,  Lake  Michigan-Huron,  August  28-September  30,  1911. 


PLATE  14 


•••\+M •-.'•.-•n  77 


2RM.     6AM. 


6RM.     f2RM 


AUG. 


6,1910 


AUG. 7, 1910 


A 


573.00 


Observed  elevation 
Barometric  effect 
Wind  effect 
Corrected  elevation 


2.90 


2.80 
2.70 


2.60 
2.50 
2:40 
2.30 
2.20 


\\ 


2.00 


\\X 


\/ 


V 


I-.80 


Observed  and  corrected  elevations  of  water  surface,  Buffalo,  August  4-7,  1910; 
barometric  effects  and  wind  effects. 


12  P.M.      6A.M. 


N. 


PLATE  15 
6PM.  12  RM.          6A.M. 


N. 


6PM.      12  P.M. 


Observed  elevation 

Barometric  effect 

"WiTidT  effect" 

Corrected  elevation 


Observed  and  corrected  elevations  of  water  surface,  Buffalo,  October  21-22,  1909; 
barometric  effects  and  wind  effects. 


J2RM.6AH 


'H. 


PLATE  16 

6RM.  12  RM.  6A.M. 


6  RM.        12  RMT. 


— * Observed  elevation 

Barometric  e-ffect 

Wind  effect 

Corrected  elevation 


Observed  and  corrected  elevations  of  water  surface,  Buffalo,  October  26-27,  1910, 
and  Cleveland,  October  27,  1910;  also  barometric  effects  and  wind  effects. 


QC 


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405  Hilgard  Avenue,  Los  Angeles,  CA  90024-1388 

Return  this  material  to  the  library 

from  which  it  was  borrowed. 


MAR  8    1993 


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